Recent zbMATH articles in MSC 58https://zbmath.org/atom/cc/582021-11-25T18:46:10.358925ZWerkzeugA singular mathematical promenadehttps://zbmath.org/1472.000012021-11-25T18:46:10.358925Z"Ghys, Étienne"https://zbmath.org/authors/?q=ai:ghys.etienneAt the first look, one may feel that the book title is a little bit strange. The word singular in the title refers to the concept of singularity of a curve and does not mean a trip made by an individual person. It is a promenade into the mathematical world. The tour is interesting, entertaining and enjoyable, but it may be little bit difficult for those who have insufficient mathematical knowledge. So some mathematical maturity is required to fully appreciate the beauty presented by the author. When you go through the subjects of it you will find it a wonderfully crafted book. The book consists of 30 chapters. Each chapter provides a rich read. Several chapters are fairly independent from the rest of the book. It is a remarkable achievement in terms of its content, structure, and style. In almost all chapters the author shows excellent examples of mathematical exposition and utilize history to enrich a contemporary mathematical investigations. Actually he weaves historical stories in between the combinatorics, complex analysis, and algebraic geometry \dots etc. and does it all in a very readable and remarkable way. The design of the book is amazing: it contains many pictures and illustrations, scanned manuscripts, references, remarks, all written in the right margin of the pages (so one has the information immediately available). The text contains many historical quotations in different languages, with translations, and interesting analysis of the mathematics of our ``classics'' (Newton, Gauss, Hipparchus \dots etc). Hence the book will please any budding or professional mathematician. I can say that, principally, for professional readers, the book is an enjoyable reading due to the versatility of subjects using too many illustrations and remarks that enriched the concepts of the classical notions. In fact most of the material in the book can be regarded as an advanced undergraduate/early graduate level, even there are some material that is significantly more advanced. One very remarkable aspects of the book is the treat of historical matters. Some of the very classical notions such as the fundamental theorem of algebra, the theory of Puisseux series, the linking number of knots, discrete mathematics, operads, resolution of curve singularities, complex singularities, and more, have been discussed and explained in an enlightening way.
The author of the book, Professor Étienne Ghys, Director of Research at the École Normale Superiere de Lyon, is a skilled, gifted versatile expositor mathematician. He wrote his book in a relaxed, informal manner with lots of exclamation marks, figures, supporting computer graphics and illustrations that are mathematically helpful and visually engaging. It is interesting to know that most of illustrations have been produced by Ghys himself and who has waived all copyright and related or neighboring rights which is a good evidence of Ghys's service towards the dissemination of mathematical ideas. Ghys is a prominent researcher, broadly in geometry and dynamics. He was awarded the Clay Award for Dissemination of Mathematics in 2015.
As the author mentioned in his book, the motivation for writing such an interesting book came from a fact brought to his attention by his colleague, Maxim Kontsevich, in 2009 that relates the relative position of the graphs of four real polynomials under certain conditions imposed on the polynomials. So he begins the book with an attractive theorem of Maxim Kontsevich scribbled for him on a Paris metro ticket who stated it in a nice: Theorem. There do not exist four polynomials \(P_1, \dots , P_4 \in R[x] \) with \(P_1(x) < P_2(x) < P_3(x) < P_4(x)\) for all small negative \(x\), and \(P_2(x) < P_4(x) < P_1(x) < P_3(x)\) for small positive \(x\).
In fact Ghys begins his promenade with this attractive theorem. Amazingly, this result basically characterizes what can or cannot happen for crossings, not only for graphs of arbitrary collections of polynomials, but indeed for all real analytic planar curves. Actually the book explores very different questions related to this problem, and follows on different ramifications. Ghys discussed the more general singularities of algebraic curves in the plane, explaining how the concepts were developed historically. I recommend to assign parts of it as an independent studies for both undergraduate and graduate students.Book review of: G.-M. Greuel et al., Singular algebraic curveshttps://zbmath.org/1472.000122021-11-25T18:46:10.358925Z"Degtyarev, Alex"https://zbmath.org/authors/?q=ai:degtyarev.alexReview of [Zbl 1411.14001].On framed simple purely real Hurwitz numbershttps://zbmath.org/1472.050072021-11-25T18:46:10.358925Z"Kazarian, M. E."https://zbmath.org/authors/?q=ai:kazaryan.maxim-e"Lando, S. K."https://zbmath.org/authors/?q=ai:lando.sergei-k"Natanzon, S. M."https://zbmath.org/authors/?q=ai:natanzon.sergei-mThe distribution relation and inverse function theorem in arithmetic geometryhttps://zbmath.org/1472.112002021-11-25T18:46:10.358925Z"Matsuzawa, Yohsuke"https://zbmath.org/authors/?q=ai:matsuzawa.yohsuke"Silverman, Joseph H."https://zbmath.org/authors/?q=ai:silverman.joseph-hillelThe paper presents an arithmetic distribution relation as well as two versions of the inverse function theorem in terms of an arithmetic distance function. In sections 3-5, the field \(K\) denotes a field with a complete set of absolute values \(\mathcal{M}_K\) satisfying a product formula. If we fix an algebraic closure \(\bar{K}\) of \(K\), an \(\mathcal{M}_K\)-constant will be a function \(\gamma \colon \mathcal{M}_{\bar{K}} \longrightarrow \mathbb{R}_{\geq 0}\) such that \(\gamma(v)\) depends only on the restriction \(v|_K\) and the set \(\{ v|_K \, | \, \gamma(v) \neq 0 \}\) is finite. The notation \(O(\mathcal{M}_K)\) will be used to denote a relation that holds up to an \(\mathcal{M}_K\)-constant. For instance \(f \leq g +O(h) + O(\mathcal{M}_K)\) means that there exist a \(C>0\) and an \(\mathcal{M}_K\)-constant \(\gamma\) such that \(f \leq g + C|h| + \gamma\).
Definition: (Local height functions) Let \(K\) be a field with a set of absolute values \(\mathcal{M}_K\). Let \(V\) be a projective variety (not necessarily irreducible) over \(K\) and \(X \subset V\) a closed subscheme. A local height is a function \(\lambda_X \colon V(\bar{K}) \times \mathcal{M}_K \longrightarrow \mathbb{R} \cup \infty\) determined by the following properties:
\begin{enumerate}
\item[(1)] If \(D\) is an effective divisor, we get the usual local height, i.e., \(\lambda_X = \lambda_D\).
\item[(2)] if \(X,X'\) are subschemes \(\lambda_{X \cap X'} = \min{(\lambda_X,\lambda_{X'})}\) (where \(X \cap X'\) denotes the subscheme with ideal sheaf .\(\mathcal{I}_X+\mathcal{I}_{X'}\) ).
\end{enumerate}
and having also many other nice properties:
\begin{enumerate}
\item[(3)] (functoriality) If \(\varphi \colon V \longrightarrow W\) denotes a morphism of varieties and \(X\) is a closed subscheme of \(W\), we have the equality: \[\lambda_{\varphi^{-1} W,V} = \lambda_{X,W}.\]
\item[(4)] Local height functions are bounded below, so up a \(\mathcal{M}_K\)-constant, we can assume that \(\lambda_X \geq 0\).
\end{enumerate}
Remark: Using the local height function associated to the boundary divisor, the local height function machinery can be extended to quasi-projective varieties.
Definition: Let \(\Delta(V)\) denotes the diagonal subvariety in \(V \times V\). The arithmetic distance function on \(V\) is the local height \[\delta_V = \lambda_{\Delta(V)}.\] It is well-defined up to an \(\mathcal{M}_K\)-bounded function and satisfies many nice properties as well, for example:
\begin{enumerate}
\item[(1)] \(\delta(P,R) \geq \min(\delta(P,Q),\delta(Q,R))\)
\item[(2)] \(\lambda_X(Q) \geq \min(\lambda_X(P),\delta(P,Q))\)
\end{enumerate}
Defintion: Let \(\varphi \colon W \longrightarrow V\) be a finite flat morphism between schemes of finite type over a field \(k\). Let \(k'\) be an algebraically closed field containing \(k\). For \(x \in W(k')\), define the multiplicity of \(\varphi\) at \(x\) by the formula \[e_\varphi(x)= \text{length}_{\mathcal{O}_{W_{k'},x}} \mathcal{O}_{W_{k'},x}/\varphi^{-1} m_{\varphi(x)} \mathcal{O}_{W_{k'},x}.\]
For finite flat morphisms \(W \longrightarrow V\), the distribution inequality bounds the arithmetic distance in the target variety in terms of the arithmetic distance of the pre-images in \(W\). In good cases, it will give a distribution relation.
Theorem: (Arithmetic distribution relation/inequality) Let \(\varphi \colon W \longrightarrow V\) be a generically étale finite flat morphism between quasi-projective geometrically integral varieties over \(K\). For all \((P,q,v) \in W(\bar{K}) \times V(\bar{K}) \times v\), we have \[\delta_V(\varphi(P),q,v) \leq \sum_{Q \in W(\bar{K}),\, \varphi(Q)=q} e_\varphi(Q) \delta_W(P,Q,v)+ O(\lambda_{ \partial(V \times W)}(P,q,v))+ O(\mathcal{M}_K).\] We say that we have an arithmetic distribution relation when the above inequality becomes an equality. For example, assuming that \(V,W\) are both smooth, we have an arithmetic relation in the following two situations:
\begin{enumerate}
\item[(1)] The map \(\varphi \colon W \longrightarrow V\) is étale (where there is not ramification).
\item[(2)] The dimensions \(\dim(V)=\dim(W)=1\) (where the ramification divisor is at most zero-dimensional).
\end{enumerate}
Remark: The above inequality does not always became an arithmetic relation. Even when we take a Galois cover \(\varphi \colon W \longrightarrow V\) with Galois group \(\text{Gal}(V/W)=\{\tau_1,\dots,\tau_n\}\), we have an inclusion of associated sheaves of ideals \[\mathcal{I}(\sum_{i=1}^n (1\times \tau_i)^* \Delta(W)) \subset \mathcal{I}((\varphi \times \varphi)^*(\Delta(V))\] that does not always gives an equality of closed schemes. Take for example \(\varphi \colon \mathbb{P}^1 \times \mathbb{P}^1 \longrightarrow \mathbb{P}^1 \times \mathbb{P}^1 \) defined by \(\varphi([x,y],[z,w])=([x^2,y^2],[z,w])\).
Remark: The above inequality can be used to obtain a quantitative inverse theorem, namely, given a finite map, how far apart from the ramification locus and the boundary we need to be, to be able to define a local inverse. Also, the inverse obtained can be shown to behave nicely with respect to the distance functions. In some sense, the distance between points is close to the distance between the inverses.
Theorem: (Inverse function theorem version \(1\)) Suppose that \(V\) and \(W\) are quasi-projective geometrically integral varieties defined over \(K\). Assume that the map \(\varphi \colon W \longrightarrow V\) is a generically étale finite flat surjective morphism of degree \(d\) also defined over K. Let us denote by \(\text{Ann}(\Omega_{W/V})\), the annihilator ideal sheaf of \(\Omega_{W/V}\) and by \(A(\varphi) \subset W\) the closed subscheme defined by \(\text{Ann}(\Omega_{W/V})\).
\begin{enumerate}
\item[(a)] There exist constants \(C_2,C_3\) and \(\mathcal{M}_K\)-constants \(C_4,C_5\) such that the following holds:\\
If the triple \((P,q,v) \in W(K) \times V(K) \times M(K)\) satisfies \[\delta_V (\varphi(P),q;v) \geq d \lambda_{A(\varphi)}(P;v) + C_2 \lambda_{ \partial_{W \times V} }(P,q;v) + C_4(v)\] then there exists a point \(Q \in W(K)\) satisfying \(\varphi(Q) = q\) and \[\delta_W(P,Q;v) \geq \delta_V (\varphi(P),q;v) -(d-1)\lambda_{A(\varphi)}(P;v)- C_3 \lambda_{\partial(W \times V )}(P, q; v) - C_5(v).\]
\item[(b)]If we take \(C_4\) to be an appropriate positive real number, instead of an \(\mathcal{M}_K\) constant, and if we also assume that \(P \notin A(\varphi)\), then the point \(Q\) in (a) is unique.
\end{enumerate}
Remark: The arithmetic distribution relation and the inverse function theorem have been used to study integral points in the following situations:
\begin{enumerate}
\item[(1)] To find uniform height estimates while working with the étale map \([n] \colon A \longrightarrow A\) on an Abelian scheme \(A \longrightarrow T\) over a base variety \(T\).
\item[(2)] To find an analogous of Siegel's theorem while working with Iterates \(f^n\) of a rational map \(f \colon \mathbb{P}^1 \longrightarrow \mathbb{P}^1\) of degree at least two.
\end{enumerate}
The first version (version \(1\)) of the inverse function theorem works simultaneously over several places. In the next version the authors present a stronger result working only over a complete field \(K\). By working over a complete field, the exponents or coefficients of the inverse function theorem are improved from \((d,d-1)\) to \((2,1)\).
Theorem: (Inverse function theorem version \(2\)). Let \( (K, | . |)\) be a complete field. Let \(W, V\) be smooth quasi-projective varieties over \(K\), and let \(\varphi \colon W \longrightarrow V\) be a generically finite generically étale morphism. Let \(E \subset W\) be the closed subscheme defined by the \(0\)-th fitting ideal sheaf of \(\Omega_{W/V}\). Fix arithmetic distance functions \(\delta_W\), \(\delta_V\), a local height function \(\lambda_E\), and a boundary function \(\lambda_{\partial V}\). Let \(B \subset W(K)\) be a bounded subset. Then there are constants \(C_{36}, C_{37}, C_{38}, C_{39} > 0\) and a bounded subset \(\tilde{B} \subset W (K)\) containing \(B\) such that for all \(P \in B\) and \(q \in V(K)\) satisfying \[P \notin E \quad \text{and} \quad \delta_V(\varphi(P),q) \geq 2 \lambda_E(P) + C_{36} \lambda_{\partial V}(q) + C_{37},\] there is a unique \(Q \in \tilde{B}\) satisfying \[\varphi(Q)=q \quad \text{and} \quad \delta_W(P,Q) \geq \delta_V(\varphi(P),q) - \lambda_E(P)-C_{38} \lambda_{\partial V}(q) - C_{39}.\]
Remark: Both versions of the inverse function theorem are suitable to prove results analogous to the continuity of the roots for polynomials. For example from the second version, we could get the following result. Let \( (K, | . |)\) be a complete field. Let \(D \in \mathbb{R}_{>0}\) and \(n \in \mathbb{Z}_{>0}\). Then there are positive constants \(C_{40},C_{41} > 0\) such that the following holds. Suppose that:
\begin{itemize}
\item \(f,g \in K[t]\) are monic polynomials of degree \(n\);
\item The Gauss norms \(|f | \leq D\) and \(|g| \leq D\);
\item There is an \(\alpha \in K\) such that \(f(\alpha) = 0\) and \(|f-g| \leq C_{40} |f''\alpha)|\).
\end{itemize}
Then there is \(\beta \in K\) such that \[g(\beta) = 0 \quad \text{and} \quad |\alpha - \beta||f'(\alpha)| \leq C_{41} |f - g|.\]
Remark: The second version of the inverse function theorem is based on a higher dimensional version of Newton's method. The point \(Q\), will be obtained as limit of a Cauchy sequence of points \(Q_0=P, Q_1, Q_2 \dots\).Linear maps preserving the Lorentz-cone spectrum in certain subspaces of \(M_n\)https://zbmath.org/1472.150392021-11-25T18:46:10.358925Z"Bueno, M. I."https://zbmath.org/authors/?q=ai:bueno.maribel-i|bueno.maria-isabel"Furtado, S."https://zbmath.org/authors/?q=ai:furtado.susana"Sivakumar, K. C."https://zbmath.org/authors/?q=ai:sivakumar.koratti-chengalrayanLet \(n \geq 3\) and consider the Lorentz cone \(\mathcal K = \{(x,x_n)\in \mathbb R^{n-1}\times \mathbb R : \lVert x \rVert \leq x_n \}\), where \(\lVert x\rVert\) is the 2-norm of \(x\). It is well-known that the Lorentz cone \(\mathcal K\) is self-dual, and so the eigenvalue complementarity problem associated with \(\mathcal K\) is, for a given real-valued \(n \times n\) matrix \(A\), finding a scalar \(\lambda \in \mathbb R\) and nonzero vector \(x \in \mathbb R^n\) such that \[x \in \mathcal K, \qquad (A-\lambda I_n)x \in \mathcal K, \qquad x^T(A-\lambda I_n)x = 0,\] where \(I_n\) denotes the \(n \times n\) matrix and \(x^T\) denotes the transpose of \(x\). The scalar \(\lambda\) is called \textit{Lorentz eigenvalue} of \(A\) and the set of all Lorentz eigenvalues of \(A\) is denoted \(\sigma_\mathcal{K}(A)\), and called the \textit{Lorentz cone spectrum of \(A\)}. The authors explore linear preserver problems related to the Lorentz cone spectrum of matrices. In other words, they obtain characterizations of linear maps \(\phi: \mathcal M \to \mathcal M\) such that \(\sigma_\mathcal{K}(\phi(A)) = \sigma_\mathcal{K}(A)\) for all \(A \in \mathcal M\), where \(\mathcal M\) is a given subspace of \(M_n(\mathbb R)\).
The authors call maps of the form \(\phi(A) = PAQ\) or \(\phi(A) = PA^TQ\), where \(P\) and \(Q\) are fixed matrices, \textit{standard linear maps} (most linear preserver problems in the literature conclude that the maps have this particular form; see [\textit{C.-K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)] for a general description). Lorentz cone spectrum preservers are considered on the following subspaces \(\mathcal M\): diagonal matrices, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) symmetric, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) arbitrary, the space of symmetric matrices, and the full matrix space \(M_n(\mathbb R)\). For the diagonal and block-diagonal spaces, the authors show (see Theorem 4.2) that Lorentz cone spectrum preservers of these spaces are, in fact, standard maps with \(P\) and \(Q\) invertible of a specified form.
For the symmetric and full matrix space case, the complete description of standard linear maps that preserve the Lorentz cone spectrum are provided. It seems to be an open problem as to whether or not all Lorentz cone spectrum preservers of the symmetric and full matrix spaces must in fact be standard maps.
As a byproduct of studying preservers in this paper, certain properties of the Lorentz cone spectrum (and the interplay with the usual spectrum) are developed for special classes of matrices; in particular, rank-one matrices.Modules with values in the space of all derivations of an algebrahttps://zbmath.org/1472.160412021-11-25T18:46:10.358925Z"Abbasi, H."https://zbmath.org/authors/?q=ai:abbasi.huzaifa"Haghighatdoost, GH."https://zbmath.org/authors/?q=ai:haghighatdoost.ghorbanaliSummary: In this paper, we construct a groupoid associated to a module with values in the space of all derivations of a unital algebra. More precisely, for a pair \((\mathcal{A}, \mathcal{G})\) consisting of an algebra \(\mathcal{A}\) with a unit, a module \(\mathcal{G}\) over the center \(Z(\mathcal{A})\) of \(\mathcal{A}\) together with a homomorphism of \(Z(\mathcal{A})\)-modules from \(\mathcal{G}\) to the space of all derivations \(\operatorname{Der}(\mathcal{A})\) of \(\mathcal{A}\), we associate a groupoid. We discuss on the equivalence relation induced from this groupoid.Strong Novikov conjecture for low degree cohomology and exotic group \(\mathrm{C}^\ast\)-algebrashttps://zbmath.org/1472.190052021-11-25T18:46:10.358925Z"Antonini, Paolo"https://zbmath.org/authors/?q=ai:antonini.paolo"Buss, Alcides"https://zbmath.org/authors/?q=ai:buss.alcides"Engel, Alexander"https://zbmath.org/authors/?q=ai:engel.alexander"Siebenand, Timo"https://zbmath.org/authors/?q=ai:siebenand.timoLet \(G\) be a discrete group, \(\Lambda^*(G) \subset H^*(BG;\mathbb Q)\) the subring generated by the rational cohomology classes of degree at most two, and let \(\mathrm{ch}: K_*(BG) \to H_*(BG;\mathbb Q)\) be the homological Chern character from the \(K\)-homology to the homology of the classifying space \(BG\) of \(G\).
It was shown in [\textit{B. Hanke} and \textit{T. Schick}, Geom. Dedicata 135, 119--127 (2008; Zbl 1149.19006)] that if for \(h\in K_*(BG)\) there exists \(c \in\Lambda^*(G)\) with \(\langle c, \mathrm{ch}(h)\rangle\neq 0\) then \(h\) is not mapped to zero under the assembly map \(K_*(BG) \to K_*(C^*_{\max}(G)) \otimes\mathbb R\).
The paper under review shows that the maximal group \(C^*\)-algebra here can be replaced by a smaller one, namely, by the exotic group \(C^*\)-algebra \(C^*_\epsilon (G)\), which is obtained from the so-called minimal exact and strongly Morita compatible crossed product functor introduced in [\textit{P. Baum} et al., Ann. \(K\)-Theory 1, No. 2, 155--208 (2016; Zbl 1331.46064)].Lie theory of multiplicative tensorshttps://zbmath.org/1472.220022021-11-25T18:46:10.358925Z"Bursztyn, Henrique"https://zbmath.org/authors/?q=ai:bursztyn.henrique"Drummond, Thiago"https://zbmath.org/authors/?q=ai:drummond.thiagoAuthors' abstract: We study tensors on Lie groupoids suitably compatible with the groupoid structure, called multiplicative. Our main result gives a complete description of these objects only in terms of infinitesimal data. Special cases include the infinitesimal counterparts of multiplicative forms, multivector fields and holomorphic structures, obtained through a unifying and conceptual method. We also give a full treatment of multiplicative vector-valued forms, particularly Nijenhuis operators and related structures.An application of Cartan's equivalence method to Hirschowitz's conjecture on the formal principlehttps://zbmath.org/1472.320102021-11-25T18:46:10.358925Z"Hwang, Jun-Muk"https://zbmath.org/authors/?q=ai:hwang.jun-mukLet \(A\) be a compact complex submanifold of a complex manifold \(X\). The submanifold \(A \subset X\) is said to \textit{satisfy the formal principle} if given a compact submanifold \(\tilde{A}\) of a complex manifold \(\tilde{X}\), a formal isomorphism \(\psi: (A/X)_{\infty} \rightarrow (\tilde{A}/\tilde{X})_{\infty}\) between the formal neighbourhoods, and a positiver integer \(l\), we can find a biholomorphism \[ \Psi: (A/X)_{\mathcal{O}} \rightarrow (\tilde{A}/\tilde{X})_{\mathcal{O}} \] such that \(\Psi|_{(A/X)_l} = \psi|_{(A/X)_l}\). Here we have written \((A/X)_{l}\) for the \(l\)-th order neighbourhood and \((A/X)_{\mathcal{O}}\) for the germ of Euclidean neighbourhoods of \(A\) in \(X\).
In order to compare the germ of \(A \subset X\) with the germ of the zero section of the normal bundle \(N_{A/X}\) (a question considered by many authors, e.g., [\textit{M. Abate} et al., Adv. Math. 220, No. 2, 620--656 (2009; Zbl 1161.32011)]) the author introduces the following terminology: A vector bundle \(W\) on a compact complex manifold \(A\) is said to satisfy the formal principle if the zero section \(0_A \subset W\) satisfies the formal principle in the classical sense.
A conjecture of \textit{A. Hirschowitz} [Ann. Math. (2) 113, 501--514 (1981; Zbl 0421.32029)] states the following:
Conjecture 1.3. Let \(A \subset X\) be an unobstructed compact submanifold of a complex manifold. Assume that the normal bundle \(N_{A/X}\) is globally generated, i.e., the sequence \[ 0 \rightarrow H^0(A,N_{A/X} \otimes m_x) \rightarrow H^0(A,N_{A/X}) \rightarrow N_{A/X,x} \rightarrow 0, \] where \(m_x\) is the maximal ideal at \(x \in A\), is exact at every \(x \in A.\) Then \(A \subset X\) satisfies the formal principle.
This in turn predicts the following:
Conjecture 1.4. A globally generated vector bundle on a compact complex manifold satisfies the formal principle.
In this paper the author obtains new results in the direction of Conjectures 1.3 and 1.4 by viewing families of submanifolds on a complex manifold as a geometric structure in the sense of Cartan (equivalence method). The main result states that if the sections of the normal bundle separate points in the setting of Conjecture 1.3, then the formal principle holds for sufficiently general deformations of \(A\) in \(X\). More precisely, one main novelty of the paper is to prove statements in terms of the Douady space, for example the main result is of the following form: ``Let \(X\) be a complex manifold and let \(K\) be a subset of the Douady space satisfying certain hypotheses (here omitted for brevity). Then there exists a nowhere-dense subset \(S\) of \(K\) such that the submanifolds corresponding to any point of \(K \setminus S\) satisfies the formal principle''.
The author proceeds to give a number of applications of this result regarding globally generated vector bundles that satisfy the formal principle (towards Conjecture 1.4). In particular Conjecture 1.4 is proven for Fano manifolds.
As a further application improvements to Cartan-Fubini type extension theorems are discussed, cf. [\textit{J.-M. Hwang} and \textit{N. Mok}, J. Math. Pures Appl. (9) 80, No. 6, 563--575 (2001; Zbl 1033.32013)], as part of a program to replace difficult-to-check transcendental conditions by algebraic conditions.The stationary disc method in the unique jet determination of CR automorphismshttps://zbmath.org/1472.320182021-11-25T18:46:10.358925Z"Bertrand, Florian"https://zbmath.org/authors/?q=ai:bertrand.florianFinite jet determination of holomorphic maps of real manifolds has gained notable attention during the recent decades. Among them, we have in particular the impressive result of \textit{S. S. Chern} and \textit{J. Moser} in their celebrated work [Acta Math. 133, 219--271 (1974; Zbl 0302.32015)] showing that every holomorphic automorphism of a certain real-analytic nondegenerate hypersurface in an arbitrary complex space, which preserves some nondegenerate point \(p\) is uniquely determined by its jets of order two at this point. One finds a large amount of considerable works on the finite jet determination of holomorphic maps in the real analytic setting.
In the smooth case, most results on this issue rely on the method of complete differential systems initiated by Cartan and Chern-Moser. But in the case of \textit{finitely smooth} manifolds, it seems that the method of stationary discs is the only known way to treat the problem of finite jet determination. Attached to a given real submanifold \(M\subset\mathbb{C}^N\), the stationary discs are actually a family of analytic invariant objects which admit a lift with a pole of order at most 1 at the origin. The main idea behind attaching analytic stationary discs to real submanifolds is a boundary value problem, namely a nonlinear problem of Riemann-Hilbert type.
The paper under review studies the finite jet determination of holomorphic automorphisms of some specific, but interesting, (degenerate and nondegenerate) kinds of real submanifolds in arbitrary complex spaces by constructing the attached stationary discs and analyzing their corresponding geometric features.Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implementedhttps://zbmath.org/1472.340042021-11-25T18:46:10.358925Z"Bérard, Pierre"https://zbmath.org/authors/?q=ai:berard.pierre-h"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernardThis paper gives an account of results emanating from Sturm's seminal work on eigenvalues and eigenfunctions of the Dirichlet problem
\[
-y''(x)+q(x)y(x) = \lambda y(x)\text{ in }]0,1[,\ y(0)=y(1)=0.
\]
The authors are particularly interested in the statement that the number of the zeros of nontrivial real linear combinations of the first \(n\) real eigenfunctions is bounded by \(n-1\), where \(n\) is any positive integer. They present different proofs, interwoven with detailed accounts of the history of the presented approaches. This historical overview also briefly discusses the generalization to the Laplace operator \(-\Delta \) with Dirichlet boundary conditions on a bounded domain in \(\mathbb{R}^n\) and counterexamples thereof.
First, the authors present Liouville's proof. Then they give an alternative proof using the strategy proposed by Gelfand, considering the first eigenfunction of the \(n\)-particle Hamiltonian restricted to Fermions. This apprears to be the first complete proof following Gelfand's suggestion. Finally, Gelfand's strategy is compared with Kellog's approach and the resulting approach via oscillation kernels.
This paper is highly recommended to anyone interested in the history of Stum-Liouville theory or working in oscillation theory.Nonlinear problems with lack of compactnesshttps://zbmath.org/1472.350052021-11-25T18:46:10.358925Z"Molica Bisci, Giovanni"https://zbmath.org/authors/?q=ai:molica-bisci.giovanni"Pucci, Patrizia"https://zbmath.org/authors/?q=ai:pucci.patriziaThis book presents recent research results on nonlinear problems with lack of compactness via critical point theory, obtained by several mathematicians. The topics covered include several nonlinear problems in the Euclidean setting as well as variational problems on manifolds. It is divided into three parts. In the first part of the book, the authors deal with the existence of solutions for quasilinear elliptic equations in \(\mathbb{R}^{N}\), involving general operators with nonstandard growth, as well as critical nonlinearities.
In the second part of the book, the existence of multiple solutions has been treated via a group-theoretical invariance in the Hilbertian framework for different problems such as the one-parameter critical elliptic equation in \(\mathbb{R}^{N}\), the scalar field equation settled on a strip-like domain of the Euclidean space \(\mathbb{R}^{N}\) and elliptic equations on the unit sphere \(S^{N}\rightarrow\mathbb{R}^{N+1}\), with \(N\geq 2\), endowed by the induced Riemannian metric and involving a possibly critical nonlinear term. In these problems, the settings are responsible for the loss of compactness.
The third part of the book is dedicated to non-compact problems arising from geometry. More precisely, the authors deal with subelliptic problems on Carnot groups and treat elliptic problems on homogeneous Hadamard manifolds, i.e. Riemannian manifolds which are complete, simply connected, with everywhere non-positive sectional curvature and with a transitive group isometries. Moreover, by taking the advantage of the intrinsic nature of the hyperbolic geometry, elliptic problems on the Poincaré ball model are studied. Finally, the authors give a celebrated principle of symmetric criticality of R. Palais intensively used along Parts II and III of the book.Periodic solutions to nonlinear Euler-Bernoulli beam equationshttps://zbmath.org/1472.350222021-11-25T18:46:10.358925Z"Chen, Bochao"https://zbmath.org/authors/?q=ai:chen.bochao"Gao, Yixian"https://zbmath.org/authors/?q=ai:gao.yixian"Li, Yong"https://zbmath.org/authors/?q=ai:li.yong.1This rather technical paper shows the existence of time periodic solutions for an Euler-Bernoulli beam equation with periodic source terms of periode \(2\pi\). The argument employed to prove this result combines a Lyapunov-Schmidt reduction to the Nash-Moser method.Periodic solutions to Klein-Gordon systems with linear couplingshttps://zbmath.org/1472.350232021-11-25T18:46:10.358925Z"Chen, Jianyi"https://zbmath.org/authors/?q=ai:chen.jianyi"Zhang, Zhitao"https://zbmath.org/authors/?q=ai:zhang.zhitao"Chang, Guijuan"https://zbmath.org/authors/?q=ai:chang.guijuan"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing.2|zhao.jing|zhao.jing.1|zhao.jing.3Summary: In this paper, we study the nonlinear Klein-Gordon systems arising from relativistic physics and quantum field theories
\[\begin{cases}
u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)=0, \\
v_{tt}\,-v_{xx}+bv+\varepsilon u+g(t,x,v)\,=0,
\end{cases}\]
where \(u,v\) satisfy the Dirichlet boundary conditions on spatial interval \([0,\pi], b>0\) and \(f,g\) are \(2\pi\)-periodic in \({t}\). We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as \(\varepsilon\) goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on \({f}\) and \({g}\), we obtain the solutions \((u_{\varepsilon},v_{\varepsilon})\) with time period \(2\pi\) for the problem as the linear coupling constant \(\varepsilon\) is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as \(\varepsilon\to 0, (u_{\varepsilon},v_{\varepsilon})\) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic solutions.Semilinear wave equation on compact Lie groupshttps://zbmath.org/1472.350672021-11-25T18:46:10.358925Z"Palmieri, Alessandro"https://zbmath.org/authors/?q=ai:palmieri.alessandroSummary: In this note, we study the semilinear wave equation with power nonlinearity \(|u|^p\) on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any \(p>1\), under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived.Blow-up for a semilinear heat equation with Fujita's critical exponent on locally finite graphshttps://zbmath.org/1472.350682021-11-25T18:46:10.358925Z"Wu, Yiting"https://zbmath.org/authors/?q=ai:wu.yitingSummary: Let \(G=(V,E)\) be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition \(CDE'(n,0)\) and uniform polynomial volume growth of degree \(m\), all non-negative solutions of the equation \(\partial_tu=\Delta u+u^{1+\alpha}\) blow up in a finite time, provided that \(\alpha =\frac{2}{m} \). We also consider the blow-up problem under certain conditions for volume growth and initial value. These results complement our previous work joined with Lin.Local regularity for the harmonic map and Yang-Mills heat flowshttps://zbmath.org/1472.350752021-11-25T18:46:10.358925Z"Afuni, Ahmad"https://zbmath.org/authors/?q=ai:afuni.ahmadSummary: We establish new local regularity results for the harmonic map and Yang-Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by \textit{K. Ecker} [Calc. Var. Partial Differ. Equ. 23, No. 1, 67--81 (2005; Zbl 1119.35026)] and the author [ibid. 55, No. 1, Paper No. 13, 14 p. (2016; Zbl 1338.35020); Adv. Calc. Var. 12, No. 2, 135--156 (2019; Zbl 1415.58016)].On the dual geometry of Laplacian eigenfunctionshttps://zbmath.org/1472.351012021-11-25T18:46:10.358925Z"Cloninger, Alexander"https://zbmath.org/authors/?q=ai:cloninger.alexander"Steinerberger, Stefan"https://zbmath.org/authors/?q=ai:steinerberger.stefanSummary: We discuss the geometry of Laplacian eigenfunctions \(-\Delta\phi=\lambda\phi\) on compact manifolds \((M, g)\) and combinatorial graphs \(G=(V,E)\). The ``dual'' geometry of Laplacian eigenfunctions is well understood on \(\mathbb{T}^d\) (identified with \(\mathbb{Z}^d)\) and \(\mathbb{R}^n\) (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of ``similarity'' \(\alpha(\phi_\lambda,\phi_\mu)\) between eigenfunctions \(\phi_\lambda\) and \(\phi_\mu\) is given by a global average of local correlations
\[
\alpha(\phi_\lambda,\phi_\mu)^2=||\phi_\lambda\phi_\mu||^{-2}_{L^2}\int_M\left(\int_Mp(t,x,y)(\phi_\lambda(y)-\phi_\lambda(x))(\phi_\mu(y)-\phi_\mu(x))dy\right)^2dx,
\]
where \(p(t,x,y)\) is the classical heat kernel and \(e^{-t\lambda}+e^{-t\mu}=1\). This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.Large energy bubble solutions for Schrödinger equation with supercritical growthhttps://zbmath.org/1472.351092021-11-25T18:46:10.358925Z"Guo, Yuxia"https://zbmath.org/authors/?q=ai:guo.yuxia"Liu, Ting"https://zbmath.org/authors/?q=ai:liu.tingSummary: We consider the following nonlinear Schrödinger equation involving supercritical growth:
\[\begin{cases}
-\Delta u+V(y)u=Q(y)u^{2^*-1+ \varepsilon}\quad \text{in }\mathbb{R}^N,\\
u>0,\quad u\in H^1(\mathbb{R}^N),\end{cases},\tag{\(\ast\)}\]
where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(N\geq 5\), and \(V(y)\) and \(Q(y)\) are bounded nonnegative functions in \(\mathbb{R}^N \). By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions \(V\) and \(Q\), we prove that for \(\varepsilon>0\) is small enough, problem \((*)\) has large number of bubble solutions whose functional energy is in the order \(\varepsilon^{-\frac{N-4}{(N-2)^2}}.\)Multi-bump standing waves for nonlinear Schrödinger equations with a general nonlinearity: the topological effect of potential wellshttps://zbmath.org/1472.351102021-11-25T18:46:10.358925Z"Jin, Sangdon"https://zbmath.org/authors/?q=ai:jin.sangdonSummary: In this article, we are interested in multi-bump solutions of the singularly perturbed problem
\[-\varepsilon^2\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^N.\]
Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities \(f\) satisfying the Berestycki-Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as \(\varepsilon\rightarrow 0\). Examples of potential wells include the following: the union of two compact smooth submanifolds of \(\mathbb{R}^N\) where these two submanifolds meet at the origin and an embedded topological submanifold of \(\mathbb{R}^N \).The Dirichlet problem on almost Hermitian manifoldshttps://zbmath.org/1472.351312021-11-25T18:46:10.358925Z"Li, Chang"https://zbmath.org/authors/?q=ai:li.chang"Zheng, Tao"https://zbmath.org/authors/?q=ai:zheng.taoSummary: We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge-Ampère type equation and of the degenerate Monge-Ampère equation.The Oseen-Frank energy functional on manifoldshttps://zbmath.org/1472.351422021-11-25T18:46:10.358925Z"Hong, Min-Chun"https://zbmath.org/authors/?q=ai:hong.minchunSummary: We observe that for a unit tangent vector field \(u\in TM\) on a 3-dimensional Riemannian manifold \(M\), there is a unique unit cotangent vector field \(A\in T^\ast M\) associated to \(u\) such that we can define the curl of \(u\) by \(dA\). Through a unit cotangent vector field \(A\in T^\ast M\), we define the Oseen-Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen-Frank energy on 3-dimensional Riemannian manifolds.Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient typehttps://zbmath.org/1472.351462021-11-25T18:46:10.358925Z"Candela, Anna Maria"https://zbmath.org/authors/?q=ai:candela.anna-maria"Salvatore, Addolorata"https://zbmath.org/authors/?q=ai:salvatore.addolorata"Sportelli, Caterina"https://zbmath.org/authors/?q=ai:sportelli.caterinaSummary: The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type
\[\begin{cases}
-\operatorname{div}(A(x,u)|\nabla u|^{p_{1}-2}\nabla u)+\frac{1}{p_1}A_u(x,u)|\nabla u|^{p_1} =G_u( x,u,v) & \text{in }{\Omega} , \\
-\operatorname{div}(B(x,v)|\nabla v|^{p_2-2}\nabla v)+\frac{1}{ p_2}B_v(x,v)|\nabla v|^{p_2} =G_v(x,u,v) &\text{in }{\Omega} , \\ u=v=0 \quad &\text{on }\partial \Omega,
\end{cases}\tag{P}\]
where \(\Omega\subset\mathbb{R}^N\) is an open bounded domain, \(p_1, p_2>1\) and \(A(x,u), B(x,v)\) are \(\mathcal{C}^1 \)-Carathéodory functions on \(\Omega\times\mathbb{R}\) with partial derivatives \(A_u(x,u)\), respectively \(B_v(x,v)\), while \(G_u(x,u,v)\), \(G_v(x,u,v)\) are given Carathéodory maps defined on \(\Omega\times\mathbb{R}\times\mathbb{R}\) which are partial derivatives of a function \(G(x,u,v)\). We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional \(\mathcal{J} \), related to problem (P), admits at least one critical point in the ``right'' Banach space \(X\). Moreover, if \(\mathcal{J}\) is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ``good'' decomposition of the Banach space \(X\) and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.Corrigendum to: ``A Morse-Smale index theorem for indefinite elliptic systems and bifurcation''https://zbmath.org/1472.351542021-11-25T18:46:10.358925Z"Portaluri, Alessandro"https://zbmath.org/authors/?q=ai:portaluri.alessandro"Waterstraat, Nils"https://zbmath.org/authors/?q=ai:waterstraat.nilsSummary: We discussed in a previous paper [ibid. 258, No. 5, 1715--1748 (2015; Zbl 1310.35111)] elliptic systems of partial differential equations on star-shaped domains and introduced the notions of conjugate radius and bifurcation radius. We proved that every bifurcation radius is a conjugate radius, and believed to have shown by an example that on the other hand not every conjugate radius is a bifurcation radius. This note reveals that our previous example was wrong, but it also introduces an improved example that shows the assertion that we claimed before.Existence of the eigenvalues for the cone degenerate \(p\)-Laplacianhttps://zbmath.org/1472.351882021-11-25T18:46:10.358925Z"Chen, Hua"https://zbmath.org/authors/?q=ai:chen.hua"Wei, Yawei"https://zbmath.org/authors/?q=ai:wei.yaweiSummary: The present paper is concerned with the eigenvalue problem for cone degenerate \(p\)-Laplacian. First the authors introduce the corresponding weighted Sobolev s-paces with important inequalities and embedding properties. Then by adapting Lusternik-Schnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.On Courant's nodal domain property for linear combinations of eigenfunctions. IIhttps://zbmath.org/1472.352502021-11-25T18:46:10.358925Z"Bérard, Pierre"https://zbmath.org/authors/?q=ai:berard.pierre-h"Helffer, Bernard"https://zbmath.org/authors/?q=ai:helffer.bernardSummary: Generalizing Courant's nodal domain theorem, the ``Extended Courant property'' is the statement that a linear combination of the first neigenfunctions has at most nnodal domains. In a previous paper [Doc. Math. 23, 1561--1585 (2018; Zbl 1403.35192)], we gave simple counterexamples to this property, including convex domains. In the present paper, using some input from numerical computations, we pursue the investigation of the Extended Courant property with two new examples, the equilateral rhombus and the regular hexagon.
For the entire collection see [Zbl 1468.35004].Upper bounds for Steklov eigenvalues of submanifolds In Euclidean space via the intersection indexhttps://zbmath.org/1472.352522021-11-25T18:46:10.358925Z"Colbois, Bruno"https://zbmath.org/authors/?q=ai:colbois.bruno"Gittins, Katie"https://zbmath.org/authors/?q=ai:gittins.katieSummary: We obtain upper bounds for the Steklov eigenvalues \(\sigma_k(M)\) of a smooth, compact, \(n\)-dimensional submanifold \(M\) of Euclidean space with boundary \(\Sigma\) that involve the intersection indices of \(M\) and of \(\Sigma\). One of our main results is an explicit upper bound in terms of the intersection index of \(\Sigma\), the volume of \(\Sigma\) and the volume of \(M\) as well as dimensional constants. By also taking the injectivity radius of \(\Sigma\) into account, we obtain an upper bound that has the optimal exponent of \(k\) with respect to the asymptotics of the Steklov eigenvalues as \(k\to\infty\).Quasimode, eigenfunction and spectral projection bounds for Schrödinger operators on manifolds with critically singular potentialshttps://zbmath.org/1472.352572021-11-25T18:46:10.358925Z"Blair, Matthew D."https://zbmath.org/authors/?q=ai:blair.matthew-d"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannick"Sogge, Christopher D."https://zbmath.org/authors/?q=ai:sogge.christopher-dSummary: We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, \(H_V=-\Delta_g+V(x)\), on compact Riemannian manifolds \((M,g)\) of dimension \(n\geq 2\), which extend the results of the third author [J. Funct. Anal. 77, No. 1, 123--138 (1988; Zbl 0641.46011)] corresponding to the case where \(V\equiv 0\). We are able to handle critically singular potentials and consequently assume that \(V\in L^{\frac{n}{2}}(M)\) and/or \(V\in\mathcal{K}(M)\) (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where \(V\equiv 0\) that go back to the third author [loc. cit.] as well as ones which arose in the work of \textit{C. E. Kenig} et al. [Duke Math. J. 55, 329--347 (1987; Zbl 0644.35012)] in the study of ``uniform Sobolev estimates'' in \(\mathbb{R}^n\). We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural \(L^p\rightarrow L^p\) spectral multiplier theorems under the assumption that \(V\in L^{\frac{n}{2}}(M)\cap\mathcal{K}(M)\). Moreover, we can also obtain natural analogs of the original \textit{R. S. Strichartz} estimates [Duke Math. J. 44, 705--714 (1977; Zbl 0372.35001)] for solutions of \((\partial_t^2-\Delta +V)u=0\). We also are able to obtain analogous results in \(\mathbb{R}^n\) and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in \(\mathbb{R}^n\) (e.g., \textit{A. D. Ionescu} and \textit{D. Jerison} [Geom. Funct. Anal. 13, No. 5, 1029--1081 (2003; Zbl 1055.35098)]; \textit{D. Jerison} and \textit{C. E. Kenig} [Ann. Math. (2) 121, 463--494 (1985; Zbl 0593.35119)]; \textit{C. E. Kenig} and \textit{N. Nadirashvili} [Math. Res. Lett. 7, No. 5--6, 625--630 (2000; Zbl 0973.35064)]; \textit{H. Koch} and \textit{D. Tataru} [J. Reine Angew. Math. 542, 133--146 (2002; Zbl 1222.35050)]; \textit{I. Rodnianski} and \textit{W. Schlag} [Invent. Math. 155, No. 3, 451--513 (2004; Zbl 1063.35035)]).First Robin eigenvalue of the \(p\)-Laplacian on Riemannian manifoldshttps://zbmath.org/1472.352622021-11-25T18:46:10.358925Z"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolong"Wang, Kui"https://zbmath.org/authors/?q=ai:wang.kuiThe purpose of this paper is to study the first Robin eigenvalue of the \(p\)-Laplacian on compact Riemannian manifolds with boundary. The authors establish Cheng's eigenvalue comparison theorem (Theorem 1.1) and sharp bounds for the first Robin eigenvalue of the \(p\)-Laplacian (Theorem 1.4).
The authors denote \((M^n, g)\) an \(n\)-dimensional smooth compact Riemannian manifold with smooth boundary \(\partial M \not= \emptyset \). Let \(\Delta_p\) denote the \(p\)-Laplacian defined for \(1 < p < \infty \) by
\[
\Delta_p u := \mbox{div}(|\nabla u|^{p-2}\nabla u)\,,
\]
for \(u \in W^{1,p}(M)\). The authors consider the following eigenvalue problem with Robin boundary condition
\[
\begin{cases}
-\Delta_p v = \lambda |v|^{p-2}v,\quad\mbox{in}\;\; M, \cr \frac{\partial v}{\partial \nu} \,|\!\nabla v|^{p-2} + \alpha|v|^{p-2} v = 0, \quad\mbox{on}\;\; \partial M,
\end{cases}\tag{1}
\]
where \(\nu\) denotes the outward unit normal vector field along \(\partial M\) and \(\alpha \in \mathbb{R} \) is called the Robin parameter. The first Robin eigenvalue for \(\Delta_p\), denoted by \(\lambda_p(M, \alpha)\), is the smallest number such that (1) admits a weak solution in the distributional sense.
Theorem 1.1. Let \(M^n(\kappa )\) denote the simply-connected \(n\)-dimensional space form with constant sectional curvature \(\kappa\) and let \(V(\kappa , R)\) be a geodesic ball of radius \(R\) in \(M^n(\kappa)\). Let \(B_R(x_0) \subset M\) be the geodesic ball of radius \(R\) centered at \(x_0\). (We always have \(R < \frac{\pi}{\sqrt{\kappa}}\) if \(\kappa > 0\) in view of the Myers theorem).
(1) Suppose \, \(\mathrm{Ric}\ge (n - 1)\kappa\) on \(B_R(x_0)\). Then \[\lambda_p(B_R(x_0), \alpha) \le \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha > 0,\]
\[\lambda_p(B_R(x_0), \alpha) \ge \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha < 0. \]
(2) Let \(\Omega \subset B_R(x_0)\) be a domain with smooth boundary. Suppose \(\mathrm{Sect} \le \kappa\) on \(\Omega\) and \(R\) is less than the injectivity radius at \(x_0\). Then \[\lambda_p(\Omega , \alpha) \ge \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha > 0,\]
\[\lambda_p(\Omega , \alpha) \le \lambda_p(V(\kappa , R), \alpha),\text{ if }\alpha < 0.\] Moreover, the equality holds if and only if \(B_R(x_0)\) (or \(\Omega\) ) is isometric to \(V(\kappa , R)\).
Now let \(R\) denote the inradius of \(M\) defined by \(R = \sup\{d(x, \partial M) : x \in M\}\). Let \(C_{\kappa ,\Lambda}(t)\) be the unique solution of \[ \begin{cases} C_{\kappa ,\Lambda}'' + \kappa\, C_{\kappa ,\Lambda}(t) = 0,\cr C_{\kappa ,\Lambda}(0) = 1, \cr C_{\kappa ,\Lambda}'(0) = -\Lambda, \end{cases} \] and define \[ T\kappa ,\Lambda(t) := \frac{C_{\kappa ,\Lambda}'(t)}{C_{\kappa ,\Lambda}(t)} . \]\\
The second main theorem states the following:
Theorem 1.4. Suppose that the Ricci curvature of \(M\) is bounded from below by \((n - 1)\kappa\) and the mean curvature of \(\partial M\) is bounded from below by \((n - 1)\Lambda\) for some \(\kappa , \Lambda \in \mathbb{R}\). Then
\[\lambda_p(M, \alpha) \ge \bar{\lambda}_p ([0, R], \alpha,\text{ if }\alpha > 0,\]
\[\lambda_p(M, \alpha) \le \bar{\lambda}_p ([0, R], \alpha),\text{ if }\alpha < 0,\]
where \(\bar{\lambda}_p ([0, R], \alpha)\) is the first eigenvalue of the one-dimensional eigenvalue problem
\[ \begin{cases}
(p - 1)|\varphi'|^{p-2}\varphi'' + (n - 1)T_{\kappa ,\Lambda}|\varphi' |^{p-2}\varphi' = -\lambda |\varphi|^{p-2}\varphi, \cr |\varphi'(0)|^{p-2}\varphi'(0) = \alpha|\varphi(0)|^{p-2}\varphi(0),\cr \varphi'(R) = 0.
\end{cases} \]
Moreover, the equality occurs if and only if \((M, g)\) is a \((\kappa , \Lambda)\)-model space defined in Definition 6.1.Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponentshttps://zbmath.org/1472.353502021-11-25T18:46:10.358925Z"Chen, Jianhua"https://zbmath.org/authors/?q=ai:chen.jianhua"Huang, Xianjiu"https://zbmath.org/authors/?q=ai:huang.xianjiu"Qin, Dongdong"https://zbmath.org/authors/?q=ai:qin.dongdong"Cheng, Bitao"https://zbmath.org/authors/?q=ai:cheng.bitaoSummary: In this paper, we study the following generalized quasilinear Schrödinger equation
\[
-\operatorname{div}(\varepsilon^2g^2(u)\nabla u)+\varepsilon^2g(u) g^\prime(u)|\nabla u|^2+V(x)u=K(x)|u|^{p-2}u+|u|^{22^\ast-2}u,\quad x\in\mathbb{R}^{N},
\]
where \(N\geqslant 3\), \(\varepsilon>0\), \(4<p<22^\ast\), \(g\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^+)\), \(V\in\mathcal{C}(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^{N})\) has a positive global minimum, and \(K\in\mathcal{C}(\mathbb{R}^{N})\cap L^\infty(\mathbb{R}^N)\) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik-Schnirelmann theory, we also prove the existence of multiple solutions.Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulationhttps://zbmath.org/1472.353512021-11-25T18:46:10.358925Z"Cingolani, S."https://zbmath.org/authors/?q=ai:cingolani.silvia"Gallo, M."https://zbmath.org/authors/?q=ai:gallo.mariano|gallo.mose|gallo.mirko|gallo.michele"Tanaka, K."https://zbmath.org/authors/?q=ai:tanaka.kazunagaConformal boundary operators, \(T\)-curvatures, and conformal fractional Laplacians of odd orderhttps://zbmath.org/1472.354352021-11-25T18:46:10.358925Z"Gover, A. Rod"https://zbmath.org/authors/?q=ai:gover.ashwin-rod"Peterson, Lawrence J."https://zbmath.org/authors/?q=ai:peterson.lawrence-jSummary: We construct continuously parametrised families of conformally invariant boundary operators on densities. These generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with conformal Laplacian power operators, the boundary operators yield conformally invariant fractional Laplacian pseudodifferential operators on the boundary of a conformal manifold with boundary. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson \(Q\)-curvature and include higher-order generalisations of the mean curvature and the \(T\)-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We discuss applications of our results and techniques in the setting of Poincaré-Einstein manifolds and also use our constructions to shed light on some conjectures of Juhl.Corrigenda to: ``Codomain rigidity of the Dirichlet to Neumann operator for the Riemannian wave equation''https://zbmath.org/1472.354582021-11-25T18:46:10.358925Z"Milne, Tristan"https://zbmath.org/authors/?q=ai:milne.tristan"Mansouri, Abdol-Reza"https://zbmath.org/authors/?q=ai:mansouri.abdol-rezaSummary: The proof of Lemma 4.4 in our article [ibid. 371, No. 12, 8781--8810 (2019; Zbl 1419.35251)] contains a flaw. In proving the existence of a minimizer of the map \(\mathbf{A} \mapsto I_\epsilon [\mathbf{A}]\) defined therein, we stated that this map is a convex function of \(\mathbf{A} \). This is incorrect, as \(I_\epsilon\) is a composition of two convex functions, a quadratic form and an absolute value, and since the absolute value function is not monotonic, there is no guarantee that the resulting functional is convex. This short article corrects this flaw by showing that there is a continuous convex functional \(J_\epsilon\) such that \(I_\epsilon [\mathbf{A}] = J_\epsilon [\mathbf{A}^2]\), and then employing weak lower semi-continuity of \(J_\epsilon\) to demonstrate the existence of a minimizer of \(I_\epsilon \).Topological mixing of Weyl chamber flowshttps://zbmath.org/1472.370122021-11-25T18:46:10.358925Z"Dang, Nguyen-Thi"https://zbmath.org/authors/?q=ai:dang.nguyen-thi"Glorieux, Olivier"https://zbmath.org/authors/?q=ai:glorieux.olivierSummary: In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.Corrigendum to: ``Differentiating the stochastic entropy in negatively curved spaces under conformal changes''https://zbmath.org/1472.370422021-11-25T18:46:10.358925Z"Ledrappier, François"https://zbmath.org/authors/?q=ai:ledrappier.francois"Shu, Lin"https://zbmath.org/authors/?q=ai:shu.linSummary: We correct an error in our work [ibid. 67, No. 3, 1115--1183 (2017; Zbl 1382.37033), Proposition 2.16], and explain the subsequent changes in the proof of the main results.On compact subsets of Sobolev spaces on manifoldshttps://zbmath.org/1472.460382021-11-25T18:46:10.358925Z"Skrzypczak, Leszek"https://zbmath.org/authors/?q=ai:skrzypczak.leszek"Tintarev, Cyril"https://zbmath.org/authors/?q=ai:tintarev.kyrilSummary: The paper considers compactness of Sobolev embeddings of non-compact manifolds, restricted to subsets (typically subspaces) defined either by conditions of symmetry (or quasisymmetry) relative to actions of compact groups, or by restriction in the number of variables, i.e. consisting of functions of the form \(f\circ \varphi\) with a fixed \(\varphi \). The manifolds are assumed to satisfy general common conditions under which Sobolev embeddings exist. We provide sufficient conditions for compactness of the embeddings, which in many situations are also necessary.The index theorem for Toeplitz operators as a corollary of Bott periodicityhttps://zbmath.org/1472.470092021-11-25T18:46:10.358925Z"Baum, Paul F."https://zbmath.org/authors/?q=ai:baum.paul-f"van Erp, Erik"https://zbmath.org/authors/?q=ai:van-erp.erikSummary: This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel's theorem [\textit{L. Boutet de Monvel}, Invent. Math. 50, 249--272 (1979; Zbl 0398.47018)]. We prove Boutet de Monvel's theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.Erratum to: ``Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming''https://zbmath.org/1472.490302021-11-25T18:46:10.358925Z"Jennane, Mohsine"https://zbmath.org/authors/?q=ai:jennane.mohsine"Kalmoun, El Mostafa"https://zbmath.org/authors/?q=ai:kalmoun.el-mostafa"Lafhim, Lahoussine"https://zbmath.org/authors/?q=ai:lafhim.lahoussineSummary: This note corrects an error in our paper [ibid. 55, No. 1, 1--11 (2021; Zbl 1468.49016)] as we should drop the expression ``\textit{with at least one strict inequality}'' in the definition of interval order in Section 2. Instead of proposing this short amendment, \textit{N. A. Gadhi} and \textit{A. Ichatouhane} [ibid. 55, No. 1, 13--22 (2021; Zbl 1468.49014)] gave a proposition that requires an additional condition on the constraint functions. However, we claim that all the results of our paper are correct once the modification above is done.An optimal control problem of forward-backward stochastic Volterra integral equations with state constraintshttps://zbmath.org/1472.490402021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmeng"Xiao, Xinling"https://zbmath.org/authors/?q=ai:xiao.xinlingSummary: This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequalities. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.Tilings with congruent edge coronaehttps://zbmath.org/1472.520262021-11-25T18:46:10.358925Z"Tomenes, Mark D."https://zbmath.org/authors/?q=ai:tomenes.mark-d"De Las Peñas, Ma. Louise Antonette N."https://zbmath.org/authors/?q=ai:de-las-penas.ma-louise-antonette-nA normal tiling of the Euclidean plane is a cover of the plane by non-overlapping closed topological discs such that their diameters are bounded from above and their inradii are bounded from below by positive constants, and such that the intersection of any two tiles is either empty or a singleton or an arc. Such arcs are called edges of the tiling. A centred edge corona is composed of the centre of an edge and of all tiles having a non-empty intersection with that edge. A tiling is called edge-transitive or isotoxal if its symmetry group acts transitively on the set of all its edges. The authors show that every normal tiling with pairwise congruent centred edge coronae is isotoxal, and they classify such tilings.
For the entire collection see [Zbl 1467.52001].Aspects of differential geometry Vhttps://zbmath.org/1472.530012021-11-25T18:46:10.358925Z"Calviño-Louzao, Esteban"https://zbmath.org/authors/?q=ai:calvino-louzao.esteban"García-Río, Eduardo"https://zbmath.org/authors/?q=ai:garcia-rio.eduardo"Gilkey, Peter"https://zbmath.org/authors/?q=ai:gilkey.peter-b"Park, JeongHyeong"https://zbmath.org/authors/?q=ai:park.jeonghyeong"Vázquez-Lorenzo, Ramón"https://zbmath.org/authors/?q=ai:vazquez-lorenzo.ramonThe content of Volume V of the book under review is divided into four chapters, numbered from 16 to 19 in order to facilitate cross references with the chapters from Volumes I--IV, numbered from 1 to 3, from 4 to 8, from 9 to 11, and from 12 to 15, respectively; see [Zbl1354.53001; Zbl1354.53002; Zbl1371.53001; Zbl1419.53002]. The present volume is devoted to elliptic operator theory and its applications to differential geometry.
Chapter 16 surveys some basic results from functional analysis. The first section provides a brief review of some elementary concepts in geometry and topology. The next two sections are devoted to establishing some standard results concerning Banach and Hilbert spaces. The last section of this chapter treats the spectral theory of compact self-adjoint operators in Hilbert space.
The basics of elliptic operator theory are presented in Chapter 17. The Fourier transform, which is one of the fundamental tools in studying the partial differential equations, is the topic of the first section. The Sobolev norms on the Schwarz space in \(\mathbb{R}^m\) are defined in the second section, and basic properties of these norms are examined. Then, the setting moves to the general case of compact Riemannian manifolds; several norms on the space of smooth sections to a vector bundle are considered and the geometry of operators of Laplace type is discussed. The last section of the chapter is devoted to the spectral theory of a self-adjoint operator of Laplace type.
The aim of Chapter 18 is to investigate the elliptic complexes which naturally arise in the field of differential geometry. The first section introduces Clifford algebras and discusses operators of Dirac type. Next, the de Rham complex, the Dolbeault complex and spinors are discussed. The last section treats the Serre duality and the Kodaira vanishing Theorem for the complex Laplacian.
Chapter 19 deals with complex geometry. First, the authors provide an introduction to multivariate holomorphic geometry. Then the geometry of complex projective spaces is widely discussed. The last two sections treat Hodge manifolds and Kodaira embedding Theorem. It is shown that any compact holomorphic manifold admits a positive line bundle if and only if it embeds as a compact holomorphic submanifold of a complex projective space of some dimension.
The present volume ends with a bibliography of 78 items, short biographies of the authors, and an index. Like the previous four books, the present volume is very well written, in a clear and precise style. Certainly, this monograph will be a basic reference in the field of differential geometry.The Morse index of a minimal surfacehttps://zbmath.org/1472.530032021-11-25T18:46:10.358925Z"Chodosh, Otis"https://zbmath.org/authors/?q=ai:chodosh.otis"Maximo, Davi"https://zbmath.org/authors/?q=ai:maximo.daviThis is a survey on global properties of minimal surfaces in 3-dimensional Euclidean space.Twisted-austere submanifolds in Euclidean spacehttps://zbmath.org/1472.530322021-11-25T18:46:10.358925Z"Ivey, Thomas A."https://zbmath.org/authors/?q=ai:ivey.thomas-a"Karigiannis, Spiro"https://zbmath.org/authors/?q=ai:karigiannis.spiroThis paper studies so-called \textit{twisted-austere pairs}, given by \((M,\mu)\) with \(M^k \subset \mathbb{R}^n\) a Riemannian submanifold of dimension \(k\), \(\mu\) a 1-form on \(M\), and such that the subbundle \(N^*M + \mu \subset T^* \mathbb{R}^n \cong \mathbb{C}^n\) is a special Lagrangian submanifold. This condition imposes strong constraints on the geometry of the submanifold \(M\). More precisely: for any normal direction \(\nu\) to \(M\), the second fundamental form \(A^{\nu}\) of \(M\) along \(\nu\), and the 1-form \(\mu\) have to satisfy a certain system of coupled non-linear PDE's, called the \textit{twisted-austere equations}.
The case \(\mu=0\) is well known, and the corresponding submanifolds \(M\) are called \textit{austere}; this case is characterized by the condition that the spectrum \(\sigma(A^{\nu})\) has to satisfy \(-\sigma(A^{\nu})=\sigma(A^{\nu})\) as a set. In particular, austere submanifolds are minimal.
After reviewing the PDE's of twisted-austere pairs \((M,\mu)\), a complete classification of these pairs is given firstly for the easy case when \(M\) is totally geodesic, secondly for dimensions \(k=1,2\), and thirdly for dimension \(k=3\). For dimension \(k=1\) it is proved that \(M\) is a straight line, and for \(k=2\) it is shown that \(M\) has to be a minimal surface and \(\mu\) an harmonic \(1\)-form on \(M\).
The case \(k=3\) is the bulk of the paper, and its classification consists of a careful analysis of the twisted-austere equations. The main conclusion (Theorem 3.1.) is that either \(M^3 \subset \mathbb{R}^n\) is ruled by lines, or \(n=5\) and \(M^3 \subset \mathbb{R}^5\) is a generalized helicoid ruled by planes.The Obata equation with Robin boundary conditionhttps://zbmath.org/1472.530472021-11-25T18:46:10.358925Z"Chen, Xuezhang"https://zbmath.org/authors/?q=ai:chen.xuezhang"Lai, Mijia"https://zbmath.org/authors/?q=ai:lai.mijia"Wang, Fang"https://zbmath.org/authors/?q=ai:wang.fangFor an $n$-dimensional closed Riemannian manifold $(M,g)$ with $\operatorname{Ric}(g)\geq (n-1)g$, \textit{A. Lichnérowicz} [Géométrie des groupes de transformations. Paris: Dunod (1958; Zbl 0096.16001)] proved that the first eigenvalue of the Laplace-Beltrami operator satisfies $\lambda_1\geq n$. Moreover, Obata has shown that the equality $\lambda_1=n$ holds if and only if $(M,g)$ is isometric to the round sphere, as a consequence of the following rigidity result: $(M,g)$ admits a non-constat function $f$ satisfying $$ \nabla^2 g+fg=0 $$ if and only if $(M,g)$ is isometric to the standard sphere. The above displayed equation is known as the Obata equation.
We now assume that $M$ has non-empty boundary $\partial M$. The paper under review studies the Obata equation under the Robin boundary condition:
\[ \begin{cases} \nabla^2 g+fg=0 & \text{ in }M,\\
\frac{\partial f}{\partial \nu} + af=0&\text{ on }\partial M,
\end{cases} \]
where $\nu$ is the outward unit normal on $\partial M$ and $a$ is a non-zero constant. As a consequence, the authors obtain that, under certain conditions including $\operatorname{Ric}(g)\geq (n-1)g$, the equality $\lambda_1=n$ holds if and only if $(M,g)$ is a spherical cup (i.e., a geodesic ball in a round sphere).A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponenthttps://zbmath.org/1472.530572021-11-25T18:46:10.358925Z"Bartsch, Thomas"https://zbmath.org/authors/?q=ai:bartsch.thomas.2|bartsch.thomas.1"Xu, Tian"https://zbmath.org/authors/?q=ai:xu.tianOn a compact spin manifold \((M^m,g)\), the authors study solutions of the nonlinear Dirac equation \(D\psi=\lambda\psi+f(|\psi|)\psi+|\psi|^{\frac{2}{m-1}}\psi\) where \(\lambda\in\mathbb{R}\) and \(f=o(s^{\frac{2}{m-1}})\) as \(s\to\infty\). Such an equation is called in the paper (NLD) and is the Euler-Lagrange equation associated to some functional \(\mathcal{L}_\lambda(\psi)\). The authors show in Theorem 2.1 that if \(f\) satisfies some conditions (called \(f_1\),\(f_2\) and \(f_3\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda>0\). Also the map \(\mathbb{R}^+\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_k,\lambda_{k+1})\). On the other hand, if \(f\) satisfies some other conditions (called \(f_1\),\(f_4\) and \(f_5\) in the paper), then (NLD) has at least one energy solution \(\psi_\lambda\) for every \(\lambda\in \mathbb{R}\setminus \{\lambda_k:k\leq 0\}\). The map \(\mathbb{R}\setminus \{\lambda_k:k\leq 0\}\to \mathbb{R}^+; \lambda\to \mathcal{L}_\lambda(\psi_\lambda)\) is continuous and nonincreasing on each interval \([\lambda_{k-1},\lambda_{k})\), if \(k\geq 2\), respectively \((\lambda_{k-1},\lambda_{k})\), if \(k\leq 1\).Some properties of Dirac-Einstein bubbleshttps://zbmath.org/1472.530582021-11-25T18:46:10.358925Z"Borrelli, William"https://zbmath.org/authors/?q=ai:borrelli.william"Maalaoui, Ali"https://zbmath.org/authors/?q=ai:maalaoui.aliSummary: We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on \(\mathbb{R}^3\), which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of \(- \frac{1}{2}\)-Killing spinors on the round sphere \(\mathbb{S}^3\).A Björling representation for Jacobi fields on minimal surfaces and soap film instabilitieshttps://zbmath.org/1472.530662021-11-25T18:46:10.358925Z"Alexander, Gareth P."https://zbmath.org/authors/?q=ai:alexander.gareth-p"Machon, Thomas"https://zbmath.org/authors/?q=ai:machon.thomasSummary: We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.On constant curvature submanifolds of space formshttps://zbmath.org/1472.530712021-11-25T18:46:10.358925Z"Dajczer, M."https://zbmath.org/authors/?q=ai:dajczer.marcos"Onti, C.-R."https://zbmath.org/authors/?q=ai:onti.christos-raent"Vlachos, Th."https://zbmath.org/authors/?q=ai:vlachos.theodorosLet \(\mathbb{Q}_{\tilde{c}}^{n+p}\) be a simply connected space form. A well-known result of \textit{E. Cartan} [Bull. Soc. Math. Fr. 47, 125--160 (1920; JFM 47.0692.03)] states that an isometric immersion \(f:M_c^n\to \mathbb{Q}_{\tilde{c}}^{n+p}\) of a connected Riemannian manifold \(M_c^n\) of constant sectional curvature \(c\) has codimension \(p\geq n-1\) and if \(p=n-1\) the normal bundle is flat, provided that \(n\geq 3\) and \(c<\tilde{c}\). The dual case \(c>\tilde{c}\) was considered by \textit{J. D. Moore} [Duke Math. J. 44, 449--484 (1977; Zbl 0361.53050)] and the same conclusion was obtained under the extra assumption that \(f\) is free of weak-umbilic points.
In this short paper, the authors consider the converse of the above-mentioned results. The main result is that if the dimension of the first normal space of the immersion \(f:M_c^n\to \mathbb{Q}_{\tilde{c}}^{n+p}, n\geq 2, c\neq \tilde{c}\) is \(n-1\) (if \(c>\tilde{c}\), it is also assumed that \(f\) is free of weak-umbilic points.), then the substantial codimension of the immersion is \(p=n-1\). Examples are also provided to show that any substantial codimension is possible if the first normal space has the highest possible rank \(n\).Energy identity and necklessness for \(\alpha\)-Dirac-harmonic maps into a spherehttps://zbmath.org/1472.530792021-11-25T18:46:10.358925Z"Li, Jiayu"https://zbmath.org/authors/?q=ai:li.jiayu"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.1"Zhu, Chaona"https://zbmath.org/authors/?q=ai:zhu.chaona"Zhu, Miaomiao"https://zbmath.org/authors/?q=ai:zhu.miaomiaoSummary: Let \((\phi_{\alpha}, \psi_{\alpha})\) be a sequence of \(\alpha \)-Dirac-harmonic maps from a Riemann surface \(M\) to a compact Riemannian manifold \(N\) with uniformly bounded energy. If the target \(N\) is a sphere \(S^{K-1}\), we show that the energy identity and necklessness hold during the interior blow-up process as \(\alpha \searrow 1\) for such a sequence .Boundary value problems for Dirac-harmonic maps and their heat flowshttps://zbmath.org/1472.530802021-11-25T18:46:10.358925Z"Liu, Lei"https://zbmath.org/authors/?q=ai:liu.lei.1"Zhu, Miaomiao"https://zbmath.org/authors/?q=ai:zhu.miaomiaoSummary: Dirac-harmonic maps are critical points of an action functional that is motivated from the nonlinear \(\sigma\)-model of quantum field theory. They couple a harmonic map like field with a nonlinear spinor field. In this article, we shall discuss the latest progress on heat flow approaches for the existence of Dirac-harmonic maps under appropriate boundary conditions. Also, we discuss the refined blow-up analysis for two types of approximating Dirac-harmonic maps arising from those heat flow approaches.Mean curvature flow in asymptotically flat product spacetimeshttps://zbmath.org/1472.530992021-11-25T18:46:10.358925Z"Kröncke, Klaus"https://zbmath.org/authors/?q=ai:kroncke.klaus"Petersen, Oliver Lindblad"https://zbmath.org/authors/?q=ai:petersen.oliver-lindblad"Lubbe, Felix"https://zbmath.org/authors/?q=ai:lubbe.felix"Marxen, Tobias"https://zbmath.org/authors/?q=ai:marxen.tobias"Maurer, Wolfgang"https://zbmath.org/authors/?q=ai:maurer.wolfgang.1"Meiser, Wolfgang"https://zbmath.org/authors/?q=ai:meiser.wolfgang"Schnürer, Oliver C."https://zbmath.org/authors/?q=ai:schnurer.oliver-christian"Szabó, Áron"https://zbmath.org/authors/?q=ai:szabo.aron"Vertman, Boris"https://zbmath.org/authors/?q=ai:vertman.borisSummary: We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M \times \mathbb{R} \), where \(M\) is asymptotically flat. If the initial hypersurface \(F_0 \subset M \times \mathbb{R}\) is uniformly spacelike and asymptotic to \(M \times \{ s \}\) for some \(s \in \mathbb{R}\) at infinity, we show that a mean curvature flow starting at \(F_0\) exists for all times and converges uniformly to \(M \times \{ s\}\) as \(t \rightarrow \infty \).Type II ancient compact solutions to the Yamabe flowhttps://zbmath.org/1472.531022021-11-25T18:46:10.358925Z"Daskalopoulos, Panagiota"https://zbmath.org/authors/?q=ai:daskalopoulos.panagiota"del Pino, Manuel"https://zbmath.org/authors/?q=ai:del-pino.manuel-a"Sesum, Natasa"https://zbmath.org/authors/?q=ai:sesum.natasaThe authors construct new type II ancient compact solutions to the Yamabe flow. These solutions are rotationally symmetric and converge to a tower of two spheres as \(t\to -\infty\). The ancient solutions to the Yamabe flow are constructed by gluing two exact solutions to the rescaled equations, that is spheres, with narrow cylindrical necks. They use perturbation theory via fixed points arguments based on sharp estimates on ancient solutions of the approximated linear equation and a careful estimation of the error terms. This result can be generalized to the gluing of \(k\) spheres for any \(k\ge 2\) in such a way that the configuration of radii of the spheres was driven by a first-order Toda system as \(t\to -\infty\).Correction to: ``Distinguished \(C_p(X)\) spaces''https://zbmath.org/1472.540042021-11-25T18:46:10.358925Z"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"Kąkol, J."https://zbmath.org/authors/?q=ai:kakol.jerzy"Leiderman, A."https://zbmath.org/authors/?q=ai:leiderman.arkady-g"Saxon, S. A."https://zbmath.org/authors/?q=ai:saxon.stephen-aCorrection to the authors' paper [ibid. 115, No. 1, Paper No. 27, 18 p. (2021; Zbl 1460.54011)].Singular intersection homologyhttps://zbmath.org/1472.550012021-11-25T18:46:10.358925Z"Friedman, Greg"https://zbmath.org/authors/?q=ai:friedman.gregThe book under review is a detailed and meticulous presentation of intersection homology by singular and PL chains. An entry point for this theory is the Poincaré duality of manifolds and its failure when singularities appear. Poincaré duality is a fundamental property allowing, for instance, the introduction of the signature which turns out to be an essential invariant of bordism. In their foundational paper [Topology 19, 135--165 (1980; Zbl 0448.55004)], \textit{M. Goresky} and \textit{R. MacPherson} work with PL spaces and restore this duality for some singular PL spaces, the PL pseudomanifolds. For that they introduce a parameter, called a perversity, assigning an integer to each singular stratum. A dual perversity \(D\overline{p}\) is also associated to each perversity \(\overline{p}\). Goresky and MacPherson define the \(\overline{p}\)-intersection homology \(I^{\overline{p}}H_{k}(X)\) from a sub-complex of PL chains and obtain a nonsingular pairing between \(I^{\overline{p}}H_{k}(X)\) and \(I^{D\overline{p}}H_{n-k}(X)\), over the rationals. In their second paper, [Invent. Math. 72, 77--129 (1983; Zbl 0529.55007)], they use sheaf theory to extend their duality to topological pseudomanifolds. Several books on intersection homology using the sheaf approach exist, as those of \textit{M. Banagl} [Topological invariants of stratified spaces. Berlin: Springer (2007; Zbl 1108.55001); \textit{F. Kirwan} and \textit{J. Woolf}, An introduction to intersection homology theory. Boca Raton, FL: Chapman \& Hall/CRC (2006; Zbl 1106.55001); \textit{J.-P. Brasselet}, Introduction to intersection homology and perverse sheaves. Paper from the 27th Brazilian Mathematics Colloquium -- 27\(\degree\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 27--31, 2009. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (2009; Zbl 1173.14001)] or the recent publication of \textit{L. G. Maxim} [Intersection homology \& perverse sheaves. With applications to singularities. Cham: Springer (2019; Zbl 07105770)]. The seminar notes of [\textit{A. Borel} (ed.), Intersection cohomology. (Notes of a Seminar on Intersection Homology at the University of Bern, Switzerland, Spring 1983). Birkhäuser/Springer, Basel (1984; Zbl 0553.14002)] combine the PL and sheaf-theoretic aspects.
As mentioned at the beginning, the book under review has the particularity to present intersection homology from a singular homology point of view. Historically, this approach was initiated by \textit{H. C. King} [Topology Appl. 20, 149--160 (1985; Zbl 0568.55003)]. Along almost 800 pages, G. Friedman describes intersection (co)-homology, with its structure of products when it exists, its properties (Mayer-Vietoris sequence, Künneth formula, Poincaré duality, etc.), and some consequences such as signatures and L-classes. In particular, by using recent developments, Friedman provides a thorough treatment of intersection homology Poincaré duality via cup and cap products that completely parallels the current approach to duality on manifolds. With the two appendices and the abundance of examples, recalls and comments all along the text, this book needs few prerequisites, essentially the content of a classical Algebraic Topology course. This makes it accessible to graduate students. This approach with explanations and illustrations in no way affects the rigor of the exposition and the precision of the statements and this contribution will be a useful reference for the expert. In addition, there are some topics treated that might be of expositional interest to a broader topological audience than just those interested in singular spaces and intersection homology, as the correspondence between PL topology through triangulations vs. PL topology through PL charts or a proof of the Künneth theorem without using acyclic models. A detailed description of the chapters follows.
-- Chapter 1 is the introduction, including a presentation of the op. cit. historical papers.
-- Chapter 2 is a florilège of stratified spaces, in use in the following. Importance is given to locally cone-like spaces and more specifically to the CS sets of Siebenmann and, of course, to PL and topological pseudomanifolds. A great feature of intersection homology is its topological invariance under certain conditions on perversities; i.e., the definition of intersection homology of a CS set depends on the given stratification but in fact, this homology depends only on the underlying homeomorphism type of the space. Goresky and MacPherson provide a proof in their second paper, with sheaf theory. In the singular topological setting, the proof of King [loc. cit.] uses an intrinsic filtration present in each CS set. In this chapter, Friedman gives an exhaustive description of this key construction and uses it in Chapter 5 to establish the topological invariance.
-- In Chapter 3, the author presents intersection homology in the simplicial, the PL and the topological settings. In the case of a simplicial complex, the equivalence of the simplicial and the PL intersection homology is set up in an explicit way.
-- Chapter 4 and 5 contain the extensions of classical properties of homology to intersection homology: excision, Mayer-Vietoris sequences, universal coefficient theorems, the particular situation of a Künneth formula with a manifold factor. Here, we note the importance of Theorem 5.1.4 for many crucial proofs concerning CS sets. Chapter 5 ends with the question left open in Chapter 3: the equivalence of the singular and PL intersection homology on a PL CS set.
-- Chapter 6. The perversities introduced by Goresky and MacPherson obey to some constraints. Several works as for instance the paper of \textit{S. E. Cappell} and \textit{J. L. Shaneson} [Ann. Math. (2) 134, No. 2, 325--374 (1991; Zbl 0759.55002)], use perversities which do not satisfy the previous rules. At some point, the correspondence between the sheaf and the singular cochain presentations fails. Thus, in this chapter, Friedman develops an intersection chain theory (called non-GM) which restores this correspondence. (This notion is due, independently, to M. Saralegi-Aranguren and G. Friedman.) If the bound \(\overline{p}(S)\leq {\mathrm{codim}}(S)-2\) is respected for any singular stratum \(S\), the non-GM intersection homology coincides with the one of the previous chapters. The rest of the book is mainly concerned with this variation. The previous study of Künneth formulae is completed in general for non-GM intersection homology.
-- Chapter 7 is the implementation of cup, cap and cross products on intersection (co)-homology, not only with field coefficients but also with coefficients in a Dedekind ring. This generality leads to a greater technicality.
-- Chapter 8 is the high point of the book: a nice presentation of intersection homology Poincaré duality for pseudomanifolds, using a cap product with a fundamental intersection homology class. The homology coefficients are in a Dedekind ring and there are some ``torsion-free'' conditions. (These conditions cannot be avoided as shown by \textit{M. Goresky} and \textit{P. Siegel} [Comment. Math. Helv. 58, 96--110 (1983; Zbl 0529.55008)].) Lefschetz duality and the cup product pairing, with a thorough analysis of the torsion pairing, are also developed. A presentation of the intersection pairing on PL pseudomanifolds as in the first treatment of Goresky-MacPherson is included.
-- Chapter 9. For \(4k\)-dimensional manifolds, Poincaré duality gives a non-singular pairing on \(H^{2k}(M;{\mathbb{Q}})\). In intersection homology, the situation is more complicated since the duality involves \(I_{\overline{p}}H^{2k}\) and \(I_{D\overline{p}}H^{2k}\). Thus, a first step is to find cases where two such intersection homologies are isomorphic. Examples are given by the Witt spaces of Siegel and the IP spaces of Pardon, described here by Friedman with their main properties. He also discusses the associated signature and the corresponding characteristic homology L-classes. A survey of pseudomanifold bordism theories completes the chapter.
In Chapter 10, the author lists some suggestions for further reading. The book concludes with two appendices, one of algebraic nature (signature, projective modules) and the second one is an introduction to PL spaces.Monopole Floer homology and the spectral geometry of three-manifoldshttps://zbmath.org/1472.570322021-11-25T18:46:10.358925Z"Lin, Francesco"https://zbmath.org/authors/?q=ai:lin.francescoThere is a well-developed theory that examines the spectrum of the Laplace operator on manifolds. Less is known about the spectrum of the Laplace operator on forms. This paper uses Seiberg-Witten theory to derive an upper bound on the first eigenvalue of the Hodge Laplacian on co-exact \(1\)-forms on a wide class of \(3\)-manifolds. This result improves prior results that began with the Seiberg-Witten proof of the adjunction inequality. After establishing this upper bound, it is applied to give a new proof of an inequality first established for hyperbolic manifolds by \textit{J. F. Brock} and \textit{N. M. Dunfield} [Invent. Math. 210, No. 2, 531--558 (2017; Zbl 1379.57023)]. This is an example of a result that follows easily once one applies one key idea. In this case after establishing the standard inequality that implies compactness results for the Seiberg-Witten moduli via a Weitzenböck formula, Lin applies the Bochner formula to the form represented by the quadratic term in the first Seiberg-Witten formula. The result is a very clean and clear proof of this interesting result.Examples of contact mapping classes of infinite order in all dimensionshttps://zbmath.org/1472.570332021-11-25T18:46:10.358925Z"Gironella, Fabio"https://zbmath.org/authors/?q=ai:gironella.fabioLet \(V\) be a smooth \((2n+1)\)-manifold and \(\xi\) a contact structure on \(V\), that is, \(\xi\) is a hyperplane distribution on \(V\) satisfying a condition of complete non-integrability. In this paper the author studies the topology of the space of contactomorphisms \(\mathrm{Diff}(V,\xi)\) of the contact manifold \((V,\xi)\), comparing it with the one of the space of diffeomorphisms \(\mathrm{Diff}(V)\) of the manifold \(V\). It is known that the space \(\mathrm{Cont}(V)\) of contact structures on \(V\) plays an important role in the study of the relations between \(\mathrm{Diff}(V,\xi)\) and \(\mathrm{Diff}(V)\). Indeed, the map \(\mathrm{Diff}(V)\to\mathrm{Cont}(V)\), defined by \(\phi \mapsto \phi_*(\xi)\), is a locally-trivial fibration with fiber \(\mathrm{Diff}(V,\xi)\). This fibration induces a long exact sequence of homotopy groups \(\ldots \rightarrow \pi_{k+1}\left(\mathrm{Cont}(V)\right)\rightarrow \pi_k\left(\mathrm{Diff}(V,\xi)\right) \xrightarrow{j_*} \pi_k\left(\mathrm{Diff}(V)\right)\rightarrow \pi_k\left(\mathrm{Cont}(V)\right)\rightarrow\ldots \) where \(j_*:\pi_k\left(\mathrm{Diff}(V,\xi)\right)\rightarrow\pi_k\left(\mathrm{Diff}(V)\right)\) is the map induced on the homotopy groups by the natural inclusion \(j:\mathrm{Diff}(V,\xi)\rightarrow\mathrm{Diff}(V)\).
This paper focuses on the problem of the existence of infinite cyclic subgroups in \(\ker(j_*\vert_{\pi_{0}})\). The only known example of such a phenomenon is found in [\textit{R. E. Gompf}, Ann. Math. (2) 148, No. 2, 619--693 (1998; Zbl 0919.57012)] and [\textit{F. Ding} and \textit{H. Geiges}, Compos. Math. 146, No. 4, 1096--1112 (2010; Zbl 1209.57021)]. More precisely, Gompf argues that \(S^{2}\times S^1\), equipped with its unique (up to isotopy) tight contact structure \(\xi_{std}\), has a contact mapping class of infinite order. Then, starting from Gompf's remark, Ding and Geiges prove that \(\ker(j_*\vert_{\pi_{0}})\) and \(\pi_1(\mathrm{Con}(S^{2}\times S^1),\xi_{std})\) are actually both isomorphic to \(\mathbb{Z}\).
In this paper the author gives examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity. This is a generalization of an observation in dimension 3 by Gompf, also reused by Ding and Geiges.Recent advances in \(L^p\)-theory of homotopy operator on differential formshttps://zbmath.org/1472.580012021-11-25T18:46:10.358925Z"Ding, Shusen"https://zbmath.org/authors/?q=ai:ding.shusen"Shi, Peilin"https://zbmath.org/authors/?q=ai:shi.peilin"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.7|wang.yong.2|wang.yong.6|wang.yong.10|wang.yong|wang.yong.8|wang.yong.3|wang.yong.1|wang.yong.9|wang.yong.5Summary: The purpose of this survey paper is to present an up-to-date account of the recent advances made in the study of \(L^p\)-theory of the homotopy operator applied to differential forms. Specifically, we will discuss various local and global norm estimates for the homotopy operator \(T\) and its compositions with other operators, such as Green's operator and potential operator.Emmy Noether's theorem on invariant variational problemshttps://zbmath.org/1472.580022021-11-25T18:46:10.358925Z"Bieber, Robert"https://zbmath.org/authors/?q=ai:bieber.robert(no abstract)On lifting of 2-vector fields to \(r\)-jet prolongation of the tangent bundlehttps://zbmath.org/1472.580032021-11-25T18:46:10.358925Z"Kurek, Jan"https://zbmath.org/authors/?q=ai:kurek.jan"Mikulski, Włodzimierz"https://zbmath.org/authors/?q=ai:mikulski.wlodzimierz-mSummary: If \(m \geq 3\) and \(r \geq 1\), we prove that any natural linear operator \(A\) lifting 2-vector fields \(\Lambda \in \Gamma (\bigwedge^2 TM)\) (i.e., skew-symmetric tensor fields of type (2,0)) on \(m\)-dimensional manifolds \(M\) into 2-vector fields \(A(\Lambda)\) on \(r\)-jet prolongation \(J^rTM\) of the tangent bundle \(TM\) of \(M\) is the zero one.Poisson principal bundleshttps://zbmath.org/1472.580042021-11-25T18:46:10.358925Z"Majid, Shahn"https://zbmath.org/authors/?q=ai:majid.shahn"Williams, Liam"https://zbmath.org/authors/?q=ai:williams.liamAuthors' abstract: We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the $q$-Hopf fibration on the standard $q$-sphere. We also construct the Poisson level of the spin connection on a principal bundle.Non-commutative geometry indomitablehttps://zbmath.org/1472.580052021-11-25T18:46:10.358925Z"Lupercio, Ernesto"https://zbmath.org/authors/?q=ai:lupercio.ernestoThe paper under review is a survey on the origin and developments of non-commutative geometry.\par In 1637, R. Descartes (see his appendix \textit{La Géométrie} to the \textit{Ditxescours\, de\, la\, Méthode}) unified the commutative algebra with geometry.\par In 1925, M. Born, V. Heisenberg, and P. Jordan proposed a foundational framework for quantum mechanics. It was quantum physics and its matrix noncommutative mathematics that inspired A. Connes' introduction of non-commutative geometry in the 1980s.\par One of the main building blocks in non-commutative geometry was the study of groupoids. They generalize groups and are categories.\par Given an étale groupoid \(\mathcal{G}\), one associates to it a non-commutative algebra \(A_{\mathcal{G}}\), the convolution algebra of \(\mathcal{G}\). Examples are given.\par Gelfand duality expresses the fact that it is the same thing to have spaces as it is to have commutative algebras. The rational algebraic topology of a commutative space can be written in terms of its commutative algebra. This allows to speak of non-commutative rational topology. All the concepts that generalize depend only on the Morita equivalence class of a (possibly noncommutative) algebra.\par The most basic and classical example of a noncommutative space is the non-commutative torus, which can be obtained as the convolution algebra of an étale groupoid. Classical complex \(n\)-dimensional compact, projective Kähler toric manifolds \(X\) are defined as equivariant, projective compactifications of the complex \(n\)-dimensional torus. Classical toric geometry is generalized by replacing all the classical tori in toric geometry for non-commutative tori. One obtains non-commutative toric varieties.\par In the last section, the author points-out further directions:\par 1. Non-commutative manifolds;\par 2. The standard model of particle physics;\par 3. The Riemann Hypothesis.\par The list of References is large and well supports the results stated in this article.Twisted moduli spaces and Duistermaat-Heckman measureshttps://zbmath.org/1472.580062021-11-25T18:46:10.358925Z"Zerouali, Ahmed J."https://zbmath.org/authors/?q=ai:zerouali.ahmed-jSummary: Following Boalch-Yamakawa, Li-Bland-Ševera and Meinrenken, we consider a certain class of moduli spaces on bordered surfaces from a quasi-Hamiltonian perspective. For a given Lie group \(G\), these character varieties parametrize flat \(G\)-connections on ``twisted'' local systems, in the sense that the transition functions take values in \(G\rtimes\mathrm{Aut}(G)\). After reviewing the necessary tools to discuss twisted quasi-Hamiltonian manifolds, we construct a Duistermaat-Heckman (DH) measure on \(G\) that is invariant under the twisted conjugation action \(g\mapsto hg\kappa(h^{-1})\) for \(\kappa\in\mathrm{Aut}(G)\), and characterize it by giving a localization formula for its Fourier coefficients. We then illustrate our results by determining the DH measures of our twisted moduli spaces.State dependent Hamiltonian delay equations and Neumann one-formshttps://zbmath.org/1472.580072021-11-25T18:46:10.358925Z"Frauenfelder, Urs"https://zbmath.org/authors/?q=ai:frauenfelder.urs-adrianThe author proves critical point results for the action functionals involving Hamiltonian terms with a state dependent delay. The studied functionals are related to time dependent perturbations of the symplectic form. Results regarding Arnold type conjecture about lower bounds on the number of periodic orbits for Hamiltonian systems are given too.Torsion and linking number for a surface diffeomorphismhttps://zbmath.org/1472.580082021-11-25T18:46:10.358925Z"Florio, Anna"https://zbmath.org/authors/?q=ai:florio.annaLet \(\mathrm{Diff}^1(S)\) denote the group of \(C^1\) diffeomorphisms on a parallelizable Riemannian surface \(S\). An isotopy in \(\mathrm{Diff}^1(S)\) is an arc \((f_t)_{t\in[0,1]}\) in \(\mathrm{Diff}^1(S)\) which is continuous with respect to the weak or compact-open \(C^1\) topology on \(\mathrm{Diff}^1(S)\). Suppose \(f_0=id_S\) and \(f_1=f\). Then the isotopy \((f_t)_{t\in[0,1]}\) can be extended to any positive time as follows: for \(t\in\mathbb{R}_+\) the \(C^1\) diffeomorphism \(f_t:S\to S\) is defined as \(f_t:=f_{\{t\}}\circ f^{[t]}\), where \(\{t\}\) and \([t]\) denote the fractionary and integer part of \(t\), respectively. The extended isotopy is also denoted as \((f_t)_t\).
Fix an orientation and endow \(S\) with a Riemannian metric so that the notion of oriented angle between two non zero vectors of the same tangent space is well-defined. Then fix a reference continuous unit vector field \(X\) over \(S\) and define the function \(v((f_t)_t):TS_\ast\times\mathbb{R}\to\mathbb{R}/(2\pi\mathbb{Z})\) by
\[
(x,\xi,t)\mapsto\theta(X(f_t(x)), Df_t(x)\xi)
\]
where the notation \(\theta(u,v)\) denotes the oriented angle between two non zero vectors \(v\) and \(u\). For \(x\in S\) and \(\xi\in T_xS\setminus\{0\}\) the angle function \(v((f_t)_t )(x,\xi,\cdot)\) is continuous and thus has a continuous determination \(\tilde{v}((f_t)_t)(x,\xi,\cdot):\mathbb{R}\to\mathbb{R}\), i.e. a continuous function \(\tilde{v}((f_t)_t )(x,\xi,\cdot):\mathbb{R}\to\mathbb{R}\) such that \(\tilde{v}((f_t)_t )(x,\xi, s)\) is a measure of the oriented angle \({v}((f_t)_t )(x,\xi, s)\) for any \(s\in\mathbb{R}\). The torsion at finite time \(n\in\mathbb{N}\setminus\{0\}\) is defined by
\[
\mathrm{Torsion}_n((f_t)_t , x, \xi) := \frac{1}{n}(\tilde{v}((f_t)_t )(x,\xi,n)-\tilde{v}((f_t)_t )(x,\xi,0)).
\]
When for some \(\xi\in T_xS\setminus\{0\}\) the quantity \(\mathrm{Torsion}_n((f_t)_t,x,\xi)\) converges as \(n\to\infty\), the torsion of the orbit of \(x\) is defined by
\[
\mathrm{Torsion}((f_t)_t , x)=\lim_{n\to\infty}\mathrm{Torsion}_n((f_t)_t , x, \xi).
\]
Let \(S=\mathbb{R}^2\) be endowed with the counterclockwise orientation, and then fix the constant vector field \(X = (1, 0)\). For an isotopy \((F_t)_t\) in \(\mathrm{Diff}^1(\mathbb{R}^2)\) joining the identity to \(F_1=F\) and \(\triangle:= \{(z_1, z_2)\in\mathbb{R}^4\,:\, z_1=z_2\}\) and define the function
\[
u((F_t)_t) : (\mathbb{R}^4\setminus\triangle)\times\mathbb{R}\to\mathbb{R}/(2\pi\mathbb{Z}),\; (z_1,z_2,t)\mapsto \theta((1,0), F_t(z_2)-F_t(z_1)).
\]
Then for each \((z_1,z_2)\in\mathbb{R}^4\setminus\triangle\) the angle function \(u((F_t)_t)(z_1, z_2,\cdot)\) has a continuous determination \(\tilde{u}((F_t)_t)(z_1, z_2,\cdot):\mathbb{R}\to\mathbb{R}\). The linking number of \(z_1\) and \(z_2\) at finite time \(n\) is defined by
\[
\mathrm{Linking}_n((F_t)_t , z_1, z_2) := \frac{1}{2}(\tilde{u}((F_t)_t)(z_1, z_2, n)-\tilde{u}((F_t)_t)(z_1, z_2, 0)),
\]
and the linking number of the orbits of \(z_1\) and \(z_2\) is defined by
\[
\mathrm{Linking}((F_t)_t , z_1, z_2) := \lim_{n\to+\infty}\mathrm{Linking}_n((F_t)_t , z_1, z_2)
\]
whenever the limit exists.
As an positive answer to a question raised by [\textit{F. Béguin} and \textit{Z. R. Boubaker}, J. Math. Soc. Japan 65, No. 1, 137--168 (2013; Zbl 1268.37064)], Theorem~1.1 in the paper under review stated that for an isotopy \((F_t)_{t\in [0,1]}\) in \(\mathrm{Diff}^1(\mathbb{R}^2)\) there exists a point \(z\in [x,y]\) so that \(\mathrm{Torsion}_1((F_t)_t, z, y-x) = l\) provided that there exist two distinct points \(x,y\in\mathbb{R}^2\), \(x\ne y\) such that \(\mathrm{Linking}_1((F_t)_t, x, y)=l\in\mathbb{R}\).
For the annulus \(\mathbb{A} =\mathbb{R}/(2\pi\mathbb{Z})\times\mathbb{R}\) and an isotopy \((f_t)_{t\in [0,1]}\) in \(\mathrm{Diff}^1(\mathbb{A})\) joining \(Id_{\mathbb{A}}\) to \(f_1 = f\) which is a \(C^1\) positive twist map, Theorem 1.2 showed that \(\mathrm{Torsion}_n((f_t)_t, z, (0,1))\in (-\pi,0)\) for any \(z\in\mathbb{A}\), \((0, 1)\in T_z\mathbb{A}\setminus\{0\}\), and any \(n\in\mathbb{N}\setminus\{0\}\). As a consequence of this and Theorem~1.1, \(\mathrm{Torsion}((f_t)_t, z)\in [-\pi,0]\) for any point \(z\in\mathbb{A}\) for which the limit of the torsion exists.Harmonic maps with torsionhttps://zbmath.org/1472.580092021-11-25T18:46:10.358925Z"Branding, Volker"https://zbmath.org/authors/?q=ai:branding.volkerThis interesting paper under review is devoted to a first study of harmonic maps that are coupled to a torsion endomorphism on the target manifold. Such maps are called ``harmonic maps with torsion'' and they appear as solutions of an equation obtained by taking the standard harmonic map equation and then changing to a connection with metric torsion. The author investigates various geometric aspects of harmonic maps with torsion. First, some Bochner type formulas are established and the effects of conformal transformations on harmonic maps with torsion are studied. Next, the stability of harmonic maps with torsion is discussed. The last part of the article investigates analytic aspects of harmonic maps with torsion. It is showed that they satisfy the unique continuation property, a removable singularity theorem is proved and a Liouville type result under a small energy assumption is stated.Weakly biharmonic maps from the ball to the spherehttps://zbmath.org/1472.580102021-11-25T18:46:10.358925Z"Fardoun, Ali"https://zbmath.org/authors/?q=ai:fardoun.ali"Montaldo, S."https://zbmath.org/authors/?q=ai:montaldo.stefano"Ratto, A."https://zbmath.org/authors/?q=ai:ratto.andreaBiharmonic maps are critical point of the \textit{bienergy functional}
\[
E_2(u)=\frac{1}{2}\int_M|\tau(u)|^2 d\,v_g,
\]
where \(\tau(u)=\textrm{trace}\,\nabla d u\). In particular, harmonic maps are trivially biharmonic maps, as harmonic maps are critical points of the \textit{energy functional}
\[
E(u)=\frac{1}{2}\int_M| d u |^2 d\,v_g.
\]
Biharmonic maps that are not harmonic are called proper biharmonic maps.
Let \(B^n\) and \(S^n\) denote the \(n\)-dimensional Euclidean unit ball and sphere respectively. The authors study the family of rotationally symmetric maps \(u_a:B^n\rightarrow S^n\subset \mathbb{R}^n\times \mathbb{R}\) given by
\[
u_a(x)=\left(\sin a \,\,\frac{x}{|x|},\cos a\right),
\]
where \(a\) is a constant in \((0,\pi/2)\). In particular, the authors show (Theorem 1.1) that such maps are proper weakly biharmonic if and only if if either \( n = 5\) and \(a = \pi/3\) or \(n = 6\) and \(a = 1/2 \arccos(-4/5)\). In any of these two cases, \(u_a\) is unstable (Theorem 1.2).
The paper has three sections and it is quite self-contained, with a recollection of the necessary results on Sobolev spaces and weak solutions presented in section 2. Proofs of the main results are given in section 3.Convergence of harmonic maps between Alexandrov spaceshttps://zbmath.org/1472.580112021-11-25T18:46:10.358925Z"Huang, Jia-Cheng"https://zbmath.org/authors/?q=ai:huang.jiacheng"Wu, Guoqiang"https://zbmath.org/authors/?q=ai:wu.guoqiangThe authors show that Gromov-Hausdroff convergence of harmonic spaces preserves harmonicity (and energy). The paper is well organized into four sections, the second of which has a nicely written recollection of the main results needed to establish and prove the current results. The last two sections are dedicated to detailed proofs of the two main theorems.Triharmonic Riemannian submersions from 3-dimensional space formshttps://zbmath.org/1472.580122021-11-25T18:46:10.358925Z"Miura, Tomoya"https://zbmath.org/authors/?q=ai:miura.tomoya"Maeta, Shun"https://zbmath.org/authors/?q=ai:maeta.shun\(k\)-polyharmonic maps, as a generalization of harmonic maps, are maps between Riermannian manifolds which are critical points of the \(k\) energy \(\frac{1}{2}\int_M |(d+\delta)^k\phi|^2dv_g\). \(k\)-polyharmonic maps for \(k=2, 3\) are called biharmonic and triharmonic maps respectively. Harmonic maps are always biharmonic and triharmonic but a biharmonic map need not be triharmonic. For some recent study and progress on biharmonic maps and biharmonic submanifolds see a recent book [\textit{Y.-L. Ou} and \textit{B.-Y. Chen}, Biharmonic submanifolds and biharmonic maps in Riemannian geometry. Hackensack, NJ: World Scientific (2020; Zbl 1455.53002)] and the references therein.
For biharmonic Riemannian submersions, it was proved in [\textit{Z.-P. Wang} and \textit{Y.-L. Ou}, Math. Z. 269, No. 3--4, 917--925 (2011; Zbl 1235.53065)] that any biharmonic Riemannian submersion from a \(3\)-dimensional space form onto a surface is harmonic. The paper under review proves that this result can be generalized to the cases of triharmonic Riemannian submersions and \(f\)-biharmonic Riemannian submersions.Biharmonic maps on principal \(G\)-bundles over complete Riemannian manifolds of nonpositive Ricci curvaturehttps://zbmath.org/1472.580132021-11-25T18:46:10.358925Z"Urakawa, Hajime"https://zbmath.org/authors/?q=ai:urakawa.hajimeThis article considers principal \(G\)-bundles, equipped with a Sasaki-type metric, over a Riemannian manifold and the canonical projection \(\pi\), which is then a Riemannian submersion.
The problem investigated is to find conditions such that \(\pi\) biharmonic implies \(\pi\) harmonic.
The first theorem proved is that if the principal \(G\)-bundle is compact and has non-positive Ricci curvature then \(\pi\) biharmonic implies \(\pi\) harmonic. The reader will notice in the proof that the one-form \(\alpha\) defined by Equation (3.7) is not quite well-defined on the base manifold unless the tension field of \(\pi\) is actually basic. This should then be compared with [\textit{C. Oniciuc}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 48, No. 2, 237--248 (2002; Zbl 1061.58015)].
The second set of conditions is to assume the principal \(G\)-bundle is non-compact complete with non-positive Ricci curvature and the energy and bienergy of \(\pi\) are finite. Then \(\pi\) biharmonic implies \(\pi\) harmonic. The proof of this second theorem is really only a rehash of \textit{N. Nakauchi} et al. [Geom. Dedicata 169, 263--272 (2014; Zbl 1316.58012)].Index of equivariant Callias-type operators and invariant metrics of positive scalar curvaturehttps://zbmath.org/1472.580142021-11-25T18:46:10.358925Z"Guo, Hao"https://zbmath.org/authors/?q=ai:guo.haoFor any Lie group \(G\) acting isometrically on a manifold \(M\), the author formulates the general notion of a \(G\)-equivariant elliptic operator that is invertible outside of a \(G\)-cocompact subset of \(M\). To establish the analogue of the Rellich lemma in this setting, the author defines \(G\)-Sobolev modules from the \(G\)-action on the space of compactly supported smooth section. It is shown that \(G\)-Callias-type operators are self-adjoint, regular in the sense of Hilbert modules and hence equivariantly invertible at infinity. The paper also gives an explicit construction of \(G\)-Callias-type operator using \(K\)-theory of an equivariant Higson corona of \(M\). Finally the paper obtains an obstruction to positive scalar curvature metrics on non-cocompact manifolds as an application of the theory.4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondencehttps://zbmath.org/1472.580152021-11-25T18:46:10.358925Z"Ashwinkumar, Meer"https://zbmath.org/authors/?q=ai:ashwinkumar.meer"Png, Kee-Seng"https://zbmath.org/authors/?q=ai:png.kee-seng"Tan, Meng-Chwan"https://zbmath.org/authors/?q=ai:tan.meng-chwanSummary: We show that the four-dimensional Chern-Simons theory studied by \textit{K. Costello} et al. [ICCM Not. 6, No. 1, 120--146 (2018; Zbl 1405.81044) and ICCM Not. 6, No. 1, 46--119 (2018; Zbl 1405.81043)], is, with Nahm pole-type boundary conditions, dual to a boundary theory that is a three-dimensional analogue of Toda theory with a novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills theory on a manifold with corners, we argue that this three-dimensional Toda theory is dual to a two-dimensional topological sigma model with A-branes on the moduli space of solutions to the Bogomolny equations. This furnishes a novel 3d-2d correspondence, which, among other mathematical implications, also reveals that modules of the 3d W-algebra are modules for the quantized algebra of certain holomorphic functions on the Bogomolny moduli space.Foliated manifolds, algebraic \(K\)-theory, and a secondary invarianthttps://zbmath.org/1472.580162021-11-25T18:46:10.358925Z"Bunke, Ulrich"https://zbmath.org/authors/?q=ai:bunke.ulrichIn this paper, Bunke introduces a numerical invariant (taking values in \({\mathbb{C}}/{\mathbb{Z}}\)) associated to foliations over odd-dimensional closed spin manifolds by means of a certain transgression of the \(\hat{A}\) form. The paper is carefully written and contains a wealth of examples making it a very pleasurable read.
The foliation \(\mathcal{F}\), defined as an involutive \(\mathbb{C}\)-vector subbundle in the complexification of the tangent bundle, is assumed to be stably trivial and equipped with a stable framing \(s\). Additionally, one fixes a complex vector bundle \(V\) over the base manifold \(M\), together with a partial connection in the direction of the foliation (a partial connection means the restriction of a connection to those covariant derivatives in the direction of the foliation). The partial connection \(\nabla^I\) on \(V\) is assumed to be flat. The normal bundle to the foliation \(T_{\mathbb{C}}M/{\mathcal{F}}\) is equipped with a natural such flat partial connection, given by the Lie bracket.
Under these hypothesis, the invariant \(\rho(M,\mathcal{F},\nabla^I,s)\in {\mathbb{C}}/{\mathbb{Z}}\) is constructed by fixing some additional geometric data: an extension of the flat partial connection to an actual connection on \(V\), a Riemannian metric on \(M\), and also an extension of the natural flat partial connection on the transverse bundle to the foliation to an actual connection over \(M\). However, if the codimension of the foliation is small enough in the sense that \(2\mathrm{codim}(\mathcal{F})<\dim(M)\), then the invariant is shown to be independent of the above additional data.
The definition of invariant involves the so-called Umkehr map in differential \(K\)-theory. In particular cases, it reduces to known invariants like the (reduced) eta invariant of twisted Dirac operators on spin manifolds, the Godbillon-Vey invariant of a foliation of codimension \(1\), or Adams' \(e\)-invariant. In the last sections of the paper, the invariant is linked to a regulator map in algebraic \(K\)-theory.Spectral flow for skew-adjoint Fredholm operatorshttps://zbmath.org/1472.580172021-11-25T18:46:10.358925Z"Carey, Alan L."https://zbmath.org/authors/?q=ai:carey.alan-l"Phillips, John"https://zbmath.org/authors/?q=ai:phillips.john"Schulz-Baldes, Hermann"https://zbmath.org/authors/?q=ai:schulz-baldes.hermannIn the present paper the authors constructed a \(\mathbb{Z}_{2}\)-valued spectral flow for paths of skew-adjoint Fredholms on a real Hilbert space. First the authors defined the \(\mathbb{Z}_{2}\)-valued spectral flow associated to a straight line path in finite dimensions. The definition simply counts the number of orientation changes of the eigenfunctions at eigenvalue crossings through 0 along the path. Then the authors gave the analytic approach to the complex spectral flow for paths of self-adjoint Fredholm operators on a complex Hilbert space, this allows to show relatively directly that the \(\mathbb{Z}_{2}\)-valued spectral flow can be calculated, similarly to the complex spectral flow, as a sum of index type contributions, provided the appropriate notion of index is used. Finally an index formula is proved which connects the \(\mathbb{Z}_{2}\)-valued spectral flow of certain paths in the skew-adjoint operators on a real Hilbert space to the \(\mathbb{Z}_{2}\)-index of an associated Toeplitz operator on the complexification. At the end the authors illustrated the theory by an explicit example given by a matrix-valued shift operator which can be considered to be the analogue in real Hilbert space of the standard Toeplitz operator in the complex case. This example is the canonical non-trivial example of \(\mathbb{Z}_{2}\)-valued spectral flow.Small-time asymptotics for subelliptic Hermite functions on \(SU(2)\) and the CR spherehttps://zbmath.org/1472.580182021-11-25T18:46:10.358925Z"Campbell, Joshua"https://zbmath.org/authors/?q=ai:campbell.joshua"Melcher, Tai"https://zbmath.org/authors/?q=ai:melcher.tai\textit{J. J. Mitchell} [J. Funct. Anal. 164, No. 2, 209--248 (1999; Zbl 0928.22010)] studied the small-time behavior of Hermite functions on compact Lie groups. In particular, he demonstrated that, when written in exponential coordinates with a natural rescaling, these functions converge to the classical Euclidean Hermite polynomials. In a subsequent work [\textit{J. J. Mitchell}, J. Math. Anal. Appl. 263, No. 1, 165--181 (2001; Zbl 0997.43007)], he proved that Hermite functions on compact Riemannian manifolds, again written in exponential coordinates with appropriate rescaling, admit asymptotic expansions with a classical Hermite polynomial as the leading coefficient.
In the paper under review, the authors investigate heat kernels related to the natural subRiemannian structure on \(\mathrm{SU}(2)\simeq \S^3\) and, more generally, on higher-order odd-dimensional spheres.
More specifically, they prove that under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on \(\mathrm{SU}(2)\) converges to their analogues on the Heisenberg group at time 1. Next, they generalize these results to the CR sphere \(\S^{2d+1} \) equipped with their natural subRiemannian structure, where the limiting spaces are now the higher-dimensional Heisenberg groups. In other words, they obtain similar results as in [Mitchell, loc. cit.] for the Hermite functions on the CR spheres.A Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundaryhttps://zbmath.org/1472.580192021-11-25T18:46:10.358925Z"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.3"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yongSummary: We give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary.Points on nodal lines with given directionhttps://zbmath.org/1472.580202021-11-25T18:46:10.358925Z"Rudnick, Zeév"https://zbmath.org/authors/?q=ai:rudnick.zeev"Wigman, Igor"https://zbmath.org/authors/?q=ai:wigman.igorThis paper treats the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. Furthermore, the authors give upper bounds for the flat torus, and a computation of the expected number for arithmetic random waves is executed.
More precisely, let $\Omega$ be a planar domain with piecewise smooth boundary, and let $f$ be an eigenfunction of the Dirichlet Laplacian with eigenvalue $E$ such that $- \varDelta f = E f$. Given a direction $\zeta \in S^1$, let $N_{\zeta}(f)$ be the number of points $x$ on the nodal line $\{ x \in \Omega: \, f(x) = 0 \}$ with normal pointing in the direction $\pm \zeta$, i.e.,
\[
N_{\zeta}(f) := \# \left\{ x \in \Omega: \, f(x) =0, \,\, \frac{ \nabla f(x)}{ \Vert \nabla f(x) \Vert } = \pm \zeta \right\}.
\]
The first result below asserts an upper bound for $N_{\zeta}(f)$ with the only exceptions being when the nodal line contains a closed geodesic. It will follow as a particular case of a structure result on the set
\[
A_{\zeta}(f) := \{ x \in \Omega: \, f(x) =0, \, \langle \nabla f(x), \zeta^{\perp} \rangle = 0 \}
\]
of nodal directional points, i.e., the set of nodal points where $\nabla f$ is orthogonal to $\zeta^{\perp}$, thus it is co-linear to $\zeta$, $A_{\zeta}(f)$ contains all the singular nodal points of $f^{-1}(0)$, and could also contain certain closed geodesics in direction orthogonal to $\zeta$.
Theorem 1. Let $\zeta \in S^1$ be a direction, and $f$ be a toral eigenfunction such that $- \varDelta f = E f$ for some $E > 0$.
(i) If $\zeta$ is rational, then the set $A_{\zeta}(f)$ consists of at most $\sqrt{E}/ \pi h(\zeta)$ closed geodesics orthogonal to $\zeta$, at most $\frac{2}{\pi^2} E$ nonsingular points not lying on the geodesics, and possibly, singular points of the nodal set, where $h(\zeta)$ is the height for a rational vector.
(ii) If $\zeta$ is not rational, then the set $A_{\zeta}(f)$ consists of at most $\frac{2}{\pi^2} E$ nonsingular points, and possibly, singular points of the nodal set.
(iii) In particular, if $A_{\zeta}(f)$ does not contain a closed geodesics, then
\[
N_{\zeta}(f) \leqslant \frac{2}{\pi^2} \cdot E
\]
holds.
Next the authors compute the expected value of $N_{\zeta}$ for arithmetic random waves, cf. [\textit{F. Oravecz} et al., Ann. Inst. Fourier (Grenoble) 58, No.1, 299--335 (2008; Zbl 1153.35058)]. There are random eigenfunctions on the torus,
\[
f(x) = f_n(x) = \sum_{ \lambda \in {\mathcal E}_n } c_{\lambda} \cdot e( \langle \lambda, x \rangle )
\]
where $e(z) = e^{ 2 \pi i z}$ and $ {\mathcal E} := \{ \lambda = ( \lambda_1, \lambda_2) \in{\mathbb Z}: \, \Vert \lambda \Vert^2 = n \}$ is the set of all representations of the integer $n = \lambda_1^2 + \lambda_2^2$ as a sum of two integer squares, and $c_{\lambda}$ are standard Gaussian random variables, identically distributed and independent wave for the constraint $c_{- \lambda} = \bar{ c}_{\lambda}$, making $f_n$ real valued eigenfunctions of the Laplacian with eigenvalue $E = 4 \pi^2 n$ for every choice of the coefficients $\{ a_{\lambda} \}, \lambda \in {\mathcal E}_{\lambda}$. Let $\mu_n$ be the atomic measure on the unit circle given by
\[
\mu_n = \frac{1}{ r_2(n)} \sum_{ \lambda \in {\mathcal E}_n }\delta_{ \lambda / \sqrt{n} },
\]
where $r_2(n) := \# {\mathcal E}_n$, and let
\[
\hat{\mu}_n (k) = \frac{1}{ r_2(n)} \sum_{ \lambda = ( \lambda_1, \lambda_2) \in {\mathcal E}_n } \left( \frac{ \lambda_1 + i \lambda_2}{\sqrt{n} } \right)^k \in {\mathbb R}
\]
be its Fourier coefficients.
Theorem 2. For $\zeta = e^{ i \theta} \in S^1$, the expected value of $N_{\zeta}(f)$ for the arithmetic random wave (1.4) is
\[
{\mathbb E} [ N_{\zeta} ] = \frac{1}{ \sqrt{2} } n ( 1 + \hat{\mu}_n(4) \cdot \cos( 4 \theta) )^{1/2}.
\]
For other related works, see e.g. [\textit{M. Krishnapur} et al., Ann. Math. (2) 177, No. 2, 699--737 (2013; Zbl 1314.60101)] as to nodal length fluctuations for arithmetic random waves, and [\textit{A. Logunov}, Ann. Math. (2) 187, No. 1, 221--239 (2018; Zbl 1384.58020)] for nodal sets of Laplace eigenfunctions.Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cuspshttps://zbmath.org/1472.580212021-11-25T18:46:10.358925Z"Müller, Werner"https://zbmath.org/authors/?q=ai:muller.werner.1|muller.werner-g"Rochon, Frédéric"https://zbmath.org/authors/?q=ai:rochon.fredericThe authors start from the definition of analytic torsion, for which the celebrated Cheeger-Müller theorem was proved. As the authors point out, when the complex of cochains is defined over \(\mathbb{Z}\), then the Reidemeister torsion can be expressed in terms of the size of the torsion subgroup of the integer cohomology and the covolume of the lattice defined by the free part in the real cohomology. Thus to establish the exponential growth of torsion subgroups in cohomology, we need to compute the limiting behavior of analytic torsion via spectral methods.
The main theorem the authors prove (page 5) is quite involved (page 913, Theorem 1.1 in the text):
Theorem. If the complex irreducible representation \(\mathcal{\rho}: G\rightarrow \mathrm{GL}(V)\) satisfies \(\rho\circ \theta\not=\rho\) and if the discrete subgroup \(\Gamma\subset G\) is such that Assumption 2.2 below holds, then \[ \log(T;E, g_{X}, h_{E})=\log(\tau(\overline{X}, E, \mu_{X}))-\frac{1}{2}\log(\tau(Z,E,\mu_{Z}))-K_{\Gamma}^{\rho}c_{\rho} \] where \(c_{|\rho}\in \mathbb R\) is an explicit constant depending on \(\rho\), \(k^{\rho}_{\Gamma}\) is the number of connected components of \(Z\) on which the cohomology with values in \(E\) is non-trivial, and \(\mu_{X}\) and \(\mu_{Z}\) are cohomology bases of \(H^{*}(\overline{X},E)\) and \(H^{*}(Z,E)\) described above. Furthermore, if \(n\) is odd, then \(\tau(Z,E,\mu_{Z})=1\) and the formula simplifies to \[ \log T(X;E,g_{X}, h_{E})=\log \tau(\overline{X},E,\mu_{X})-K^{\rho}_{\Gamma} c_{\rho} \]
The main technical contribution of this theorem is that they are able to control the contribution from the degeneration of the ``cusp ends''. The authors worked out explicit examples like
\begin{itemize}
\item \(G=\mathrm{Spin}(d,1)\). In this case the above formula simplies to \[ \log T(X;E,g_{X}, h_{E})=\log \tau(\overline{X},E)-\frac{1}{2}\log \tau(Z,E) \]
\item \(G=\mathrm{Spin}(3,1)\cong \mathrm{SL}(2,\mathbb{C})\) and \(X:=\Gamma \backslash G / K\) is the complement of a hyperbolic knot. In this case the formula simplies to \[ T(X;E,g_{X},h_{E})=\tau(\overline{X},E) \]
\item \(G=\mathrm{SO}_{0}(d,1)\). For full detail we refer the reader to Corollary 1.7 of the paper. In this case the formula gives \[ \tau(\overline{X}, E_{\tau(m),\mu_{X,m})}=C_{n}\mathrm{vol}(X)m^{n(n+1)/2+1}+\mathcal{O}(m^{n(n+1)/2} \log(m)) \]
\end{itemize}
To obtain Theorem 0.1, the authors work with singular metrics of the form \[ g_{\epsilon}=\frac{dx^2}{x^2+\epsilon^2}+(x^2+\epsilon^2)g_{\partial \overline{X}}, x\in (-\delta, \delta) \] in a tubular neighborhood \(N\cong \partial \overline{X}\times (-\delta, \delta)_{x}\). The rest of the work heavily uses b-calculus from [\textit{R. B. Melrose}, The Atiyah-Patodi-Singer index theorem. Wellesley, MA: A. K. Peters, Ltd. (1993; Zbl 0796.58050)].
It should be pointed out that the method used in the paper is not the only method available to treat space with cusp type singularity. Currently, there are at least three (independent?) alternative methods available for problem of this type: The relative hyperbolic theory invented by Sarnak, Jorgenson and Rolf Lundelius; the ``relative compact perturbation theory'' invented by Siarhei Finski and other followers of Bismut; and lastly the (recent) work by \textit{G. Freixas i Montplet} and \textit{R. A. Wentworth} [Ann. Sci. Éc. Norm. Supér. (4) 52, No. 5, 1265--1303 (2019; Zbl 1440.14043); J. Differ. Geom. 115, No. 3, 475--528 (2020; Zbl 1454.58030)] on singular spaces like modular curves.
Reviewer's remark: The adoption of b-calculus to the present setting seems to be an over-kill from my naive perspective. Indeed, most of the paper (pages 923--942) are devoted to set up the b-calculus and use it to analyze the degeneration of heat kernel/resolvent in the cusp setting. While the analysis is quite involved as one needs to work through the single surgery space viewing as manifold with corners, the authors overcome a non-technical essential difficulty in Section 4 to understand the \(L^{2}\)-kernel of \(D_{b}\). To do this, the authors compute the \(L^{2}\)-cohomology of \(F_{P}(Y)\) with coefficients in \(E\) (Lemma 4.3, Proposition 4.4). The proof involves a version of the Hodge-de Rham theorem using spectral sequence of polyhomogeneous operators. From my naive perspective, this is a great achievement that they did not mention in the introduction part of their paper. This reminds me of \textit{J. I. Burgos Gil}'s work using log-singular metrics and log-log forms for Arakelov theory on Shimura varieties (for a reference, see [Lect. Notes Math. 2276, 377--401 (2021; Zbl 1458.14031)]).
What is unclear to me is whether we can by-pass the technical part of the paper using alternative methods to derive the main result. In other words, is it possible for us to ``re-interpret'' the main result of the paper from an arithmetic point of view by suggesting it may be an equality of morphisms on the moduli space of (real) arithmetic varieties? The intricate explicit equalities in statement (1.6) and (1.7) suggest there may be something deeper that can provide a unified perspective on these results.A computational study on lens spaces isospectral on formshttps://zbmath.org/1472.580222021-11-25T18:46:10.358925Z"Lauret, Emilio A."https://zbmath.org/authors/?q=ai:lauret.emilio-aAuthor's abstract: We make a computational study to know what kind of isospectralities among lens spaces and lens orbifolds exist considering the Hodge-Laplace operators acting on smooth p-forms. Several facts evidenced by the computational results are proved and some others are conjectured.The structure of Gaussian minimal bubbleshttps://zbmath.org/1472.600372021-11-25T18:46:10.358925Z"Heilman, Steven"https://zbmath.org/authors/?q=ai:heilman.steven-mSummary: It is shown that \(m\) disjoint sets with fixed Gaussian volumes that partition \(\mathbb{R}^n\) with minimum Gaussian surface area must be \((m-1)\)-dimensional. This follows from a second variation argument using infinitesimal translations. The special case \(m=3\) proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when \(m=3\), the three minimal sets are adjacent 120 degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove the \(m=4\) Triple Bubble Conjecture for the Gaussian measure. Our methods combine the Colding-Minicozzi theory of Gaussian minimal surfaces with some arguments used in the Hutchings-Morgan-Ritoré-Ros proof of the Euclidean Double Bubble Conjecture.Nonisometric surface registration via conformal Laplace-Beltrami basis pursuithttps://zbmath.org/1472.650762021-11-25T18:46:10.358925Z"Schonsheck, Stefan C."https://zbmath.org/authors/?q=ai:schonsheck.stefan-c"Bronstein, Michael M."https://zbmath.org/authors/?q=ai:bronstein.michael-m"Lai, Rongjie"https://zbmath.org/authors/?q=ai:lai.rongjieSummary: Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us to compute geometrically meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a 'target shape' and its deformed LB eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noises on the underlying manifolds and errors within the given landmarks or feature functions.Analysis and implementation of isogeometric boundary elements for electromagnetismhttps://zbmath.org/1472.780022021-11-25T18:46:10.358925Z"Wolf, Felix"https://zbmath.org/authors/?q=ai:wolf.felixPublisher's description: This book presents a comprehensive mathematical and computational approach for solving electromagnetic problems of practical relevance, such as electromagnetic scattering and the cavity problems. After an in-depth introduction to the mathematical foundations of isogeometric analysis, which discusses how to conduct higher-order simulations efficiently and without the introduction of geometrical errors, the book proves quasi-optimal approximation properties for all trace spaces of the de Rham sequence, and demonstrates inf-sup stability of the isogeometric discretisation of the electric field integral equation (EFIE). Theoretical properties and algorithms are described in detail. The algorithmic approach is, in turn, validated through a series of numerical experiments aimed at solving a set of electromagnetic scattering problems. In the last part of the book, the boundary element method is combined with a novel eigenvalue solver, a so-called contour integral method. An algorithm is presented, together with a set of successful numerical experiments, showing that the eigenvalue solver benefits from the high orders of convergence offered by the boundary element approach. Last, the resulting software, called BEMBEL (Boundary Element Method Based Engineering Library), is reviewed: the user interface is presented, while the underlying design considerations are explained in detail. Given its scope, this book bridges an important gap between numerical analysis and engineering design of electromagnetic devices.Topological pseudo entropyhttps://zbmath.org/1472.810222021-11-25T18:46:10.358925Z"Nishioka, Tatsuma"https://zbmath.org/authors/?q=ai:nishioka.tatsuma"Takayanagi, Tadashi"https://zbmath.org/authors/?q=ai:takayanagi.tadashi"Taki, Yusuke"https://zbmath.org/authors/?q=ai:taki.yusukeSummary: We introduce a pseudo entropy extension of topological entanglement entropy called topological pseudo entropy. Various examples of the topological pseudo entropies are examined in three-dimensional Chern-Simons gauge theory with Wilson loop insertions. Partition functions with knotted Wilson loops are directly related to topological pseudo (Rényi) entropies. We also show that the pseudo entropy in a certain setup is equivalent to the interface entropy in two-dimensional conformal field theories (CFTs), and leverage the equivalence to calculate the pseudo entropies in particular examples. Furthermore, we define a pseudo entropy extension of the left-right entanglement entropy in two-dimensional boundary CFTs and derive a universal formula for a pair of arbitrary boundary states. As a byproduct, we find that the topological interface entropy for rational CFTs has a contribution identical to the topological entanglement entropy on a torus.A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalitieshttps://zbmath.org/1472.810902021-11-25T18:46:10.358925Z"Antunes, Pedro R. S."https://zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao"Benguria, Rafael D."https://zbmath.org/authors/?q=ai:benguria.rafael-d"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Ourmières-Bonafos, Thomas"https://zbmath.org/authors/?q=ai:ourmieres-bonafos.thomaslet \(\Omega \subset {\mathbb R}^2\) be a \(C^\infty\) simply connected domain and let \(n = (n_1,n_2)^\top\) be the outward pointing normal field on \(\partial\Omega\). The Dirac operator with infinite mass boundary conditions in \(L^2(\Omega,{\mathbb C}^2)\) is defined as \[D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\
-2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix}, \] with domain \(\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},\) where \(\mathbf{n} := n_1 + \mathrm{i} n_2\) and \(\partial_z, \partial_{\bar{z}}\) are the Wirtinger operators. The spectrum of \(D^\Omega\) is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity \[ \cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.\] The authors prove the following estimate \[E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D}) \] with equality if and only if \(\Omega\) is a disk, where \(r_i\) is the inradius of \(\Omega\) and \(\mathbb D\) is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of \(E_1(\Omega)\). \(E > 0\) is the first non-negative eigenvalue of \(D^\Omega\) if and only if \(\mu^\Omega(E) = 0\), where \[\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.\] \par The authors propose the following conjecture \[\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0\] and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality \(E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})\) (it is still an open question).Integrable deformed \(T^{1,1}\) sigma models from 4D Chern-Simons theoryhttps://zbmath.org/1472.811162021-11-25T18:46:10.358925Z"Fukushima, Osamu"https://zbmath.org/authors/?q=ai:fukushima.osamu"Sakamoto, Jun-ichi"https://zbmath.org/authors/?q=ai:sakamoto.junichi"Yoshida, Kentaroh"https://zbmath.org/authors/?q=ai:yoshida.kentarohSummary: Recently, a variety of deformed \(T^{1,1}\) manifolds, with which 2D non-linear sigma models (NLSMs) are classically integrable, have been presented by Arutyunov, Bassi and Lacroix (ABL) [\textit{G. Arutyunov} et al., J. High Energy Phys. 2021, No. 3, Paper No. 62, 48 p. (2021; Zbl 1461.81054)]. We refer to the NLSMs with the integrable deformed \(T^{1,1}\) as the ABL model for brevity. Motivated by this progress, we consider deriving the ABL model from a 4D Chern-Simons (CS) theory with a meromorphic one-form with four double poles and six simple zeros. We specify boundary conditions in the CS theory that give rise to the ABL model and derive the sigma-model background with target-space metric and anti-symmetric two-form. Finally, we present two simple examples 1) an anisotropic \(T^{1,1}\) model and 2) a \(G / H \) \(\lambda \)-model. The latter one can be seen as a one-parameter deformation of the Guadagnini-Martellini-Mintchev model.Periodic coherent states decomposition and quantum dynamics on the flat torushttps://zbmath.org/1472.811212021-11-25T18:46:10.358925Z"Zanelli, Lorenzo"https://zbmath.org/authors/?q=ai:zanelli.lorenzoSummary: We provide a result on the coherent states decomposition for functions in \(L^2 (\mathbb{T}^n)\) where \(\mathbb{T}^n := (\mathbb{R} / 2\pi \mathbb{Z})^n\). We study such a decomposition with respect to the quantum dynamics related to semiclassical elliptic pseudodifferential operators, and we prove a related invariance result.
For the entire collection see [Zbl 1471.47002].5-dimensional Chern-Simons gauge theory on an interval: massive spin-2 theory from symmetry breaking via boundary conditionshttps://zbmath.org/1472.811712021-11-25T18:46:10.358925Z"Torabian, Mahdi"https://zbmath.org/authors/?q=ai:torabian.mahdiSummary: In this note, we revisit the 4-dimensional theory of massive gravity through compactification of an extra dimension and geometric symmetry breaking. We dimensionally reduce the 5-dimensional topological Chern-Simons gauge theory of (anti) de Sitter group on an interval. We apply non-trivial boundary conditions at the endpoints to break all of the gauge symmetries. We identify different components of the gauge connection as invertible vierbein and spin-connection to interpret it as a gravitational theory. The effective field theory in four dimensions includes the dRGT potential terms and has a tower of Kaluza-Klein states without massless graviton in the spectrum. The UV cut of the theory is the Planck scale of the 5-dimensional gravity \(l^{-1}\). If \(\zeta\) is the scale of symmetry breaking and \(L\) is the length of the interval, then the masses of the lightest graviton \(m\) and the level \(n\) (for \(n< Ll^{-1}\)) KK gravitons \(m_{\mathrm{KK}}^{(n)}\) are determined as \(m = (\zeta L^{-1})^{\frac{1}{2}} \ll m_{\mathrm{KK}}^{(n)} = n L^{-1} \). The 4-dimensional Planck mass is \(m_{\mathrm{Pl}} \sim (Ll^{-3})^{\frac{1}{2}}\) and we find the hierarchy \(\zeta < m < L^{-1} < l^{-1} < m_{\mathrm{Pl}}\).Abelian Fock-Schwinger representation for the quark propagator in external gauge fieldhttps://zbmath.org/1472.811722021-11-25T18:46:10.358925Z"Zubkov, M. A."https://zbmath.org/authors/?q=ai:zubkov.mikhail-aSummary: We develop the Diakonov-Petrov approach to the representation of the non-Abelian Wilson loop in an Abelian form. First of all, we extend this approach to the Abelian representation of the parallel transporter corresponding to the open curve rather than to the closed contour. Next, we extend this representation to the case of the non-compact groups. The obtained expressions allow us to derive the Abelian form of the Schwinger-Fock representation for the quark propagator in an external color gauge field.Nonrelativistic strings on \(R \times S^2\) and integrable systemshttps://zbmath.org/1472.811952021-11-25T18:46:10.358925Z"Roychowdhury, Dibakar"https://zbmath.org/authors/?q=ai:roychowdhury.dibakarSummary: We show that the (torsional) nonrelativistic string sigma models on \(R \times S^2\) can be mapped into \textit{deformed} Rosochatius like integrable models in one dimension. We also explore the associated Hamiltonian constrained structure by introducing appropriate Dirac brackets. These results show some solid evidence of the underlying integrable structure in the nonrelativistic sector of the gauge/string duality.Averaging over moduli in deformed WZW modelshttps://zbmath.org/1472.812062021-11-25T18:46:10.358925Z"Dong, Junkai"https://zbmath.org/authors/?q=ai:dong.junkai"Hartman, Thomas"https://zbmath.org/authors/?q=ai:hartman.thomas-e"Jiang, Yikun"https://zbmath.org/authors/?q=ai:jiang.yikunSummary: WZW models live on a moduli space parameterized by current-current deformations. The moduli space defines an ensemble of conformal field theories, which generically have \(N\) abelian conserved currents and central charge \(c > N \). We calculate the average partition function and show that it can be interpreted as a sum over 3-manifolds. This suggests that the ensemble-averaged theory has a holographic dual, generalizing recent results on Narain CFTs. The bulk theory, at the perturbative level, is identified as \( \mathrm{U}(1)^{2N}\) Chern-Simons theory coupled to additional matter fields. From a mathematical perspective, our principal result is a Siegel-Weil formula for the characters of an affine Lie algebra.Non-invertible global symmetries and completeness of the spectrumhttps://zbmath.org/1472.812322021-11-25T18:46:10.358925Z"Heidenreich, Ben"https://zbmath.org/authors/?q=ai:heidenreich.ben"McNamara, Jacob"https://zbmath.org/authors/?q=ai:mcnamara.jacob"Montero, Miguel"https://zbmath.org/authors/?q=ai:montero.miguel"Reece, Matthew"https://zbmath.org/authors/?q=ai:reece.matthew"Rudelius, Tom"https://zbmath.org/authors/?q=ai:rudelius.tom"Valenzuela, Irene"https://zbmath.org/authors/?q=ai:valenzuela.ireneSummary: It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of \textit{twist vortices}: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.Mildly flavoring domain walls in \(\mathrm{Sp} (N)\) SQCDhttps://zbmath.org/1472.812352021-11-25T18:46:10.358925Z"Benvenuti, Sergio"https://zbmath.org/authors/?q=ai:benvenuti.sergio"Spezzati, Paolo"https://zbmath.org/authors/?q=ai:spezzati.paoloSummary: We consider supersymmetric domain walls of four-dimensional \(\mathcal{N} = 1\) \(\mathrm{Sp} (N)\) SQCD with \(F = N + 1\) and \(F = N + 2\) flavors. First, we study numerically the differential equations defining the walls, classifying the solutions. When \(F = N + 2\), in the special case of the parity-invariant walls, the naive analysis does not provide all the expected solutions. We show that an infinitesimal deformation of the differential equations sheds some light on this issue. Second, we discuss the \(3d \) \(\mathcal{N} = 1\) Chern-Simons-matter theories that should describe the effective dynamics on the walls. These proposals pass various tests, including dualities and matching of the vacua of the massive \(3d\) theory with the \(4d\) analysis. However, for \(F = N +2\), the semiclassical analysis of the vacua is only partially successful, suggesting that yet-to-be-understood strong coupling phenomena are into play in our \(3d \) \(\mathcal{N} = 1\) gauge theories.Stokes phenomena in 3d \(\mathcal{N} = 2 \) \(\mathrm{SQED}_2\) and \(\mathbb{CP}^1\) modelshttps://zbmath.org/1472.812402021-11-25T18:46:10.358925Z"Jain, Dharmesh"https://zbmath.org/authors/?q=ai:jain.dharmesh"Manna, Arkajyoti"https://zbmath.org/authors/?q=ai:manna.arkajyotiSummary: We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of \(\mathbb{CP}^1\) model by considering it as a specific decoupling limit of \(\mathrm{SQED}_2\) model. This approach involves using a \(\mathbb{Z}_3\) symmetry that leaves the supersymmetric parameter space of \(\mathrm{SQED}_2\) model invariant to transform a pair of \(\mathrm{SQED}_2\) holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the \(\mathrm{SQED}_2\) model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomorphic blocks can be reduced to the known triplet of \(\mathbb{CP}^1\) blocks in a particular decoupling limit where two of the chiral multiplets in the \(\mathrm{SQED}_2\) model are made infinitely massive. This reduction then correctly reproduces the Stokes regions and matrices of the \(\mathbb{CP}^1\) blocks. Along the way, we find six pairs of \(\mathrm{SQED}_2\) holomorphic blocks in total, which lead to six Stokes-like regions covering uniquely the full parameter space of the \(\mathrm{SQED}_2\) model.Compactifying 5d superconformal field theories to 3dhttps://zbmath.org/1472.812442021-11-25T18:46:10.358925Z"Sacchi, Matteo"https://zbmath.org/authors/?q=ai:sacchi.matteo"Sela, Orr"https://zbmath.org/authors/?q=ai:sela.orr"Zafrir, Gabi"https://zbmath.org/authors/?q=ai:zafrir.gabiSummary: Building on recent progress in the study of compactifications of \(6d \) \((1, 0)\) superconformal field theories (SCFTs) on Riemann surfaces to \(4d \) \(\mathcal{N} = 1\) theories, we initiate a systematic study of compactifications of \(5d \) \(\mathcal{N} = 1\) SCFTs on Riemann surfaces to \(3d \) \(\mathcal{N} = 2\) theories. Specifically, we consider the compactification of the so-called rank 1 Seiberg \({E}_{N_f+1}\) SCFTs on tori and tubes with flux in their global symmetry, and put the resulting \(3d\) theories to various consistency checks. These include matching the (usually enhanced) IR symmetry of the \(3d\) theories with the one expected from the compactification, given by the commutant of the flux in the global symmetry of the corresponding \(5d\) SCFT, and identifying the spectrum of operators and conformal manifolds predicted by the \(5d\) picture. As the models we examine are in three dimensions, we encounter novel elements that are not present in compactifications to four dimensions, notably Chern-Simons terms and monopole superpotentials, that play an important role in our construction. The methods used in this paper can also be used for the compactification of any other \(5d\) SCFT that has a deformation leading to a \(5d\) gauge theory.Supersymmetric indices on \(I \times T^2\), elliptic genera and dualities with boundarieshttps://zbmath.org/1472.812452021-11-25T18:46:10.358925Z"Sugiyama, Katsuyuki"https://zbmath.org/authors/?q=ai:sugiyama.katsuyuki"Yoshida, Yutaka"https://zbmath.org/authors/?q=ai:yoshida.yutakaSummary: We study three dimensional \(\mathcal{N} = 2\) supersymmetric theories on \(I \times M_2\) with 2d \(\mathcal{N} = (0, 2)\) boundary conditions at the boundaries \(\partial(I \times M_2) = M_2 \sqcup M_2\), where \(M_2 = \mathbb{C}\) or \(T^2\). We introduce supersymmetric indices of three dimensional \(\mathcal{N} = 2\) theories on \(I \times T^2\) that couple to elliptic genera of 2d \(\mathcal{N} = (0, 2)\) theories at the two boundaries. We evaluate the \(I \times T^2\) indices in terms of supersymmetric localization and study dualities on the \(I \times M_2\). We consider the dimensional reduction of \(I \times T^2\) to \(I \times S^1\) and obtain the localization formula of 2d \(\mathcal{N} = (2, 2)\) supersymmetric indices on \(I \times S^1\). We illustrate computations of open string Witten indices based on gauged linear sigma models. Correlation functions of Wilson loops on \(I \times S^1\) agree with Euler pairings in the geometric phase and also agree with cylinder amplitudes for B-type boundary states of Gepner models in the Landau-Ginzburg phase.Universality of ultra-relativistic gravitational scatteringhttps://zbmath.org/1472.812482021-11-25T18:46:10.358925Z"Di Vecchia, Paolo"https://zbmath.org/authors/?q=ai:di-vecchia.paolo"Heissenberg, Carlo"https://zbmath.org/authors/?q=ai:heissenberg.carlo"Russo, Rodolfo"https://zbmath.org/authors/?q=ai:russo.rodolfo"Veneziano, Gabriele"https://zbmath.org/authors/?q=ai:veneziano.gabrieleSummary: We discuss the ultra-relativistic gravitational scattering of two massive particles at two-loop (3PM) level. We find that in this limit the real part of the eikonal, determining the deflection angle, is universal for gravitational theories in the two derivative approximation. This means that, regardless of the number of supersymmetries or the nature of the probes, the result connects smoothly with the massless case discussed since the late eighties by Amati, Ciafaloni and Veneziano. We analyse the problem both by using the analyticity and crossing properties of the scattering amplitudes and, in the case of the maximally supersymmetric theory, by explicit evaluation of the 4-point 2-loop amplitude using the results for the integrals in the full soft region. The first approach shows that the observable we are interested in is determined by the inelastic tree-level amplitude describing the emission of a graviton in the high-energy double-Regge limit, which is the origin of the universality property mentioned above. The second approach strongly suggests that the inclusion of the whole soft region is a necessary (and possibly sufficient) ingredient for recovering ultra relativistic finiteness and universality at the 3PM level. We conjecture that this universality persists at all orders in the PM expansion.Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equationshttps://zbmath.org/1472.812492021-11-25T18:46:10.358925Z"Leung, Naichung Conan"https://zbmath.org/authors/?q=ai:leung.naichung-conan"Ma, Ziming Nikolas"https://zbmath.org/authors/?q=ai:ma.ziming-nikolas"Young, Matthew B."https://zbmath.org/authors/?q=ai:young.matthew-bThe topic of the article arises to the reconstruction problem in [\textit{A. Strominger} et al., Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] mirror symmetry. Previously the notion of a scattering diagramm was investigated by [\textit{M. Kontsevich} and \textit{Y. Soibelman}, Prog. Math. 244, 321--385 (2006; Zbl 1114.14027)]. The authors develop an asymptotic analytic approach to the study of scattering diagrams. They investigate the asymptotic behavior of Maurer-Cartan elements of a (dg) Lie algebra. The authors found an alternative geometric differential approach to the proofs of the consistent completion of the scattering diagrams, previously investigated by Kontsevich-Soibelman [loc. cit.], \textit{M. Gross} and \textit{B. Siebert} [Ann. Math. (2) 174, No. 3, 1301--1428 (2011; Zbl 1266.53074)] and \textit{T. Bridgeland} [Algebr. Geom. 4, No. 5, 523--561 (2017; Zbl 1388.16013)]. The paper under reviewing deals with the geometric interpretation of theta-functions, their wall-crossing, which allow to give a combinatorial description of Hall algebra theta functions for acyclic quivers with nondegenerate skew- symmetric Euler functionsSpectral representation of lattice gluon and ghost propagators at zero temperaturehttps://zbmath.org/1472.813092021-11-25T18:46:10.358925Z"Dudal, David"https://zbmath.org/authors/?q=ai:dudal.david"Oliveira, Orlando"https://zbmath.org/authors/?q=ai:oliveira.orlando-anibal"Roelfs, Martin"https://zbmath.org/authors/?q=ai:roelfs.martin"Silva, Paulo"https://zbmath.org/authors/?q=ai:silva.paulo-roberto|silva.paulo-m-p|silva.paulo-h-d|silva.paulo-j-s|silva.paulo-f|silva.paulo-a-sSummary: We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two -- mathematically equivalent -- versions of the Källén-Lehmann spectral integral.Universal renormalization procedure for higher curvature gravities in \(D \leq 5\)https://zbmath.org/1472.830302021-11-25T18:46:10.358925Z"Araya, Ignacio J."https://zbmath.org/authors/?q=ai:araya.ignacio-j"Edelstein, José D."https://zbmath.org/authors/?q=ai:edelstein.jose-d"Rivadulla Sánchez, Alberto"https://zbmath.org/authors/?q=ai:sanchez.alberto-rivadulla"Vázquez Rodríguez, David"https://zbmath.org/authors/?q=ai:rodriguez.david-vazquez"Vilar López, Alejandro"https://zbmath.org/authors/?q=ai:lopez.alejandro-vilarSummary: We implement a universal method for renormalizing AdS gravity actions applicable to arbitrary higher curvature theories in up to five dimensions. The renormalization procedure considers the extrinsic counterterm for Einstein-AdS gravity given by the \textit{Kounterterms} scheme, but with a theory-dependent coupling constant that is fixed by the requirement of renormalization for the vacuum solution. This method is shown to work for a generic higher curvature gravity with arbitrary couplings except for a zero measure subset, which includes well-known examples where the asymptotic behavior is modified and the AdS vacua are degenerate, such as Chern-Simons gravity in 5D, Conformal Gravity in 4D and New Massive Gravity in 3D. In order to show the universality of the scheme, we perform a decomposition of the equations of motion into their normal and tangential components with respect to the Poincaré coordinate and study the Fefferman-Graham expansion of the metric. We verify the cancellation of divergences of the on-shell action and the well-posedness of the variational principle.Twistor actions for integrable systemshttps://zbmath.org/1472.830632021-11-25T18:46:10.358925Z"Penna, Robert F."https://zbmath.org/authors/?q=ai:penna.robert-fSummary: Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.Molecules as Riemannian manifoldshttps://zbmath.org/1472.923352021-11-25T18:46:10.358925Z"Rahimi, M."https://zbmath.org/authors/?q=ai:rahimi.morteza|rahimi.mehdi|rahimi.mona|rahimi.m-ostad|rahimi.mansour|rahimi.mehran|rahimi.mahboobeh|rahimi.m-y|rahimi.mohammad-reza-ostad|rahimi.mostafa|rahimi.mohamadtaghi|rahimi.mohammad-a"Abbasi, M."https://zbmath.org/authors/?q=ai:abbasi.morteza|abbasi.maziar|abbasi.malek|abbasi.molai-ali|abbasi.masoume|abbasi.mehdi|abbasi.mohsin-manshad|abbasi.mahdiyeh|abbasi.m-r|abbasi.masumeh|abbasi.m-j|abbasi.mohammad-yahya|abbasi.mokhtar|abbasi.mahboubeh|abbasi.m-h|abbasi.muhammad-athar|abbasi.muhammad-aqib|abbasi.muhammad-ali-babarSummary: In this paper, using the electronic charge densities, we assign a Riemannian manifold to any molecular system. Since we have many important quantities on a Riemannian Manifold, we may define them for molecular systems. A concept of distance on the configuration space of a molecule is defined. We also give a lower bound for the min square error of an unbiased estimator of molecular configurations.