Recent zbMATH articles in MSC 58https://zbmath.org/atom/cc/582023-11-13T18:48:18.785376ZUnknown authorWerkzeugHeat kernels, stochastic processes and functional inequalities. Abstracts from the workshop held October 30 -- November 5, 2022https://zbmath.org/1521.000182023-11-13T18:48:18.785376ZSummary: The workshop provided a forum for recent progress on a wide array of topics at the nexus of Analysis (elliptic, subelliptic and parabolic differential equations), Geometry (Riemannian and sub-Riemannian geometries, metric measure spaces, geometric analysis and curvature), and Probability Theory (Brownian motion, Dirichlet spaces, stochastic calculus and random media). The workshop provides a unique opportunity to encourage and foster interactions between mathematicians who share some common interests but might use different research tools or work in different mathematical settings.Mini-workshop: A geometric fairytale full of spectral gaps and random fruit. Abstracts from the mini-workshop held November 27 -- December 3, 2022https://zbmath.org/1521.000202023-11-13T18:48:18.785376ZSummary: In many situations, most prominently in quantum mechanics, it is important to understand well the eigenvalues and associated eigenfunctions of certain self-adjoint differential operators. The goal of this workshop was to study the strong link between spectral properties of such operators and the underlying geometry which might be randomly generated. By combining ideas and methods from spectral geometry and probability theory, we hope to stimulate new research including important topics such as Bose-Einstein condensation in random environments.Relative equivariant coarse index and relative \(L^2\)-indexhttps://zbmath.org/1521.190012023-11-13T18:48:18.785376Z"Chen, Xiaoman"https://zbmath.org/authors/?q=ai:chen.xiaoman"Liu, Yanlin"https://zbmath.org/authors/?q=ai:liu.yanlin"Zhou, Dapeng"https://zbmath.org/authors/?q=ai:zhou.dapengThe authors firstly generalize Roe's relative coarse index to the equivariant setting. Then, they prove the equivariant relative coarse index theorem. Moreover, the authors also generalizes Atiyah's \(L^{2}\)-index theorem to relative equivariant setting.
Reviewer: Tatsuki Seto (Tokyo)Band width and the Rosenberg indexhttps://zbmath.org/1521.190022023-11-13T18:48:18.785376Z"Kubota, Yosuke"https://zbmath.org/authors/?q=ai:kubota.yosukeLet \(M\) be a closed spin manifold. It is necessary to assume the injectivity of the Baum-Connes map in order to get the strongest results relating the nonexistence, stably, of a metric of positive scalar curvature on \(M\) to the higher index theory of \(M\)'s Dirac operator. The paper under review establishes a relationship between a concept closely related to positive scalar curvature and the Dirac operator without assuming the injectivity of the Baum-Connes map. One version of the relationship proven in this paper is: if \(M\) has infinite \(\mathcal K \mathcal O\)-width, the Rosenberg index of \(M\) is not zero. \(\mathcal K \mathcal O\)-width refers to the supremum over possible metrics on \(M\) of the width of bands in \(M\) that have inward boundaries on which the Dirac operators have non-vanishing higher indices. Bands are codimension-zero submanifolds with inward-facing and outward-facing boundaries. Estimates of [\textit{M. Gromov}, in: Foundations of mathematics and physics one century after Hilbert. New perspectives. Cham: Springer. 135--158 (2018; Zbl 1432.53052); Geom. Funct. Anal. 28, No. 3, 645--726 (2018; Zbl 1396.53068); in: Perspectives in scalar curvature. In 2 volumes. Singapore: World Scientific. 1--514 (2023; Zbl 07733259)], adapted to higher-index formulations by \textit{S. Cecchini} [Geom. Funct. Anal. 30, No. 5, 1183--1223 (2020; Zbl 1455.58008)], \textit{R. Zeidler} [J. Differ. Geom. 122, No. 1, 155--183 (2022; Zbl 1515.53053); SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 127, 15 p. (2020; Zbl 1455.19005)], show that infinite \(\mathcal K \mathcal O\)-width prevents \(M\) from admitting a metric of positive scalar curvature.
The author proves the result by showing that the \(K\)-theory class of the maximal equivariant coarse index of the Dirac operator on \(M\)'s universal cover maps by a homomorphism to the corresponding \(K\)-theory class on the universal cover of a band's inward boundary. By definition of the class of bands considered, the latter class is nonzero. The author's methods extend to certain noncompact manifolds and to a multiwidth generalization of band width.
Reviewer: Peter Haskell (Blacksburg)Hochschild's method for describing the Mackenzie obstruction to construction of a transitive Lie algebroidhttps://zbmath.org/1521.220022023-11-13T18:48:18.785376Z"Gasimov, V. A."https://zbmath.org/authors/?q=ai:gasimov.v-aSummary: The technique developed in Hochschild's papers ``Lie algebra kernels and cohomology'' [Am. J. Math. 76, 698--716 (1954; Zbl 0055.26601)] and ``Cohomology classes of finite type and finite dimensional kernels for Lie algebras'' [Am. J. Math. 76, 763--778 (1954; Zbl 0057.27204)] can be applied to the special case of transitive Lie algebroids. Our task is to transfer the Hochschild construction to the case of transitive Lie algebroids and prove a similar theorem.
[The name Hochschild is misspelled in the original title.]
For the entire collection see [Zbl 1478.35002].On \(L^{\infty}\) estimate for complex Hessian quotient equations on compact Kähler manifoldshttps://zbmath.org/1521.320192023-11-13T18:48:18.785376Z"Sui, Zhenan"https://zbmath.org/authors/?q=ai:sui.zhenan"Sun, Wei"https://zbmath.org/authors/?q=ai:sun.wei.10|sun.wei.1|sun.wei|sun.fei|sun.wei.2Summary: In this paper, we shall study \(L^{\infty}\) estimate for complex quotient equations on compact Kähler manifolds. The method applies to more general equations satisfying a set of structure conditions. Moreover, this method also applies to the stability estimate for complex Hessian equations.Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in \(\mathbb{C}^2\)https://zbmath.org/1521.320372023-11-13T18:48:18.785376Z"Olver, Peter J."https://zbmath.org/authors/?q=ai:olver.peter-j"Sabzevari, Masoud"https://zbmath.org/authors/?q=ai:sabzevari.masoud"Valiquette, Francis"https://zbmath.org/authors/?q=ai:valiquette.francisThis paper studies the problem of normal forms and equivalence of nondegenerate real hypersurfaces \(M\subset{\mathbb{C}^2}\) under the pseudo-group action of holomorphic transformations by the method of equivariant moving frames. The paper by \textit{S. S. Chern} and \textit{J. K. Moser} [Acta Math. 133, 219--271 (1975; Zbl 0302.32015)] is very important in the field of local geometry of real hypersurfaces in the complex space \(\mathbb{C}^n\). In [loc. cit.], Chern and Moser applied two complementary methods to study the problem: normal forms based on Taylor expansions, and the Cartan equivalence method. The two methods bring a different range of tools and results, and their precise interrelationship remains not entirely clear. This paper derives normal forms in the case of singularly umbilic points by the effective method of equivariant moving frames, which contributes to reconciling the Cartan equivalence and normal form methods. The most important contribution of the method of equivariant moving frames are the recurrence relations, which are easy to use. Firstly, the authors perform the normalizations up to order six through using the recurrence relations for the (partially) normalized invariants. Then they re-establish the Chern-Moser normal forms for, respectively, the non-umbilic and umbilic cases. Finally, they give a proof of the main theorem, where they consider nondegenerate hypersurfaces \(M\) that are singularly umbilic at a point \(p\), and the normal forms of the three cases -- generic, semi-circular, and circular are derived successively.
Reviewer: Guokuan Shao (Zhuhai)Explicit Carleman formulas for the Dolbeault cohomology in concave domainshttps://zbmath.org/1521.320402023-11-13T18:48:18.785376Z"Shestakov, Ivan V."https://zbmath.org/authors/?q=ai:shestakov.ivan-vSummary: In [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 45, 253--262 (1999; Zbl 1005.32024)] \textit{M. Nacinovich} et al. suggested an abstract method for constructing Carleman formulas for the Dolbeault complex. What has been lacking are simple and explicit examples. In this article we present a Carleman formula for Dolbeault cohomology classes given on a part of the boundary whose comlement is concave. As corollary we derive a uniqueness theorem for the Dolbeault cohomology.Variational techniques for a system of Sturm-Liouville equationshttps://zbmath.org/1521.340212023-11-13T18:48:18.785376Z"Shokooh, Saeid"https://zbmath.org/authors/?q=ai:shokooh.saeidThe paper is concerned with the sixth order Sturm-Liouville problem
\[
\begin{cases}
-\left(p_i(x)u_i'''(x)\right)'''+\left(q_i(x)u_i''(x)\right)''-\left(r_i(x)u_i'(x)\right)'+s_i(x)u_i(x) =\lambda F_{u_i}(x,u_1,\dots,u_n)\\
\text{ for } 0<x<T,\\
u_i(0)=u_i(T)=u_i''(0)=u_i''(T)=u_i^{(iv)}(0)=u_i^{(iv)}(T)=0
\end{cases} \tag{1}
\]
for \(i=1,\dots,n\), where \(n\in \mathbb{N}\), \(T>0\), \(\lambda\) is a positive parameter, the functions \(p_i,q_i,r_i,s_i\in L^{\infty}([0,T])\) with \(p_i^-:=\mathrm{ess} \inf_{x\in [0,T]}p_i(x)>0\) and
\[
\max\left\{-\frac{q_i^- T^2}{\pi^2},-\frac{q_i^- T^2}{\pi^2}-\frac{r_i^- T^4}{\pi^4},-\frac{q_i^- T^2}{\pi^2}-\frac{r_i^- T^4}{\pi^4}-\frac{s_i^- T^6}{\pi^6},\right\}<p_i^-,
\]
for any \(i=1,\dots,n\), the function \(F:[0,T]\times \mathbb{R}^n\to\mathbb{R}\) satisfies some assumptions, and \(F_{u_i}\) denotes the partial derivative of \(F\) with respect to \(u_i\) for \(i=1,\dots,n\). By using the critical point theory and variational methods, the author gives an interval for the parameter \(\lambda\) such that problem (1) has at least one nontrivial weak solution.
Reviewer: Rodica Luca (Iaşi)Existence of ground state solution for a class of one-dimensional Kirchhoff-type equations with asymptotically cubic nonlinearitieshttps://zbmath.org/1521.340282023-11-13T18:48:18.785376Z"Khoutir, Sofiane"https://zbmath.org/authors/?q=ai:khoutir.sofianeIn this paper, the author considers the following Kirchoff-type equation \[-\Big(1+\int_{\mathbb{R}} |u'|^2dx\Big)u''+p(x)u=l(x)u^3+f(x,u),~~ x\in \mathbb{R},\] where \(p,l\in C(\mathbb{R})\) and \(f\in C(\mathbb{R}\times\mathbb{R},\mathbb{R}).\) By using the Non-Nehari manifold method in combination with the Mountain Pass Theorem and concentration-compactness argument, the existence of a ground state solution to the above equation is established, in the case when the nonlinearity is asymptotically cubic with respect to the unknown function.
Reviewer: Sotiris K. Ntouyas (Ioannina)Uniqueness in weighted Lebesgue spaces for an elliptic equation with drift on manifoldshttps://zbmath.org/1521.350062023-11-13T18:48:18.785376Z"Meglioli, Giulia"https://zbmath.org/authors/?q=ai:meglioli.giulia"Roncoroni, Alberto"https://zbmath.org/authors/?q=ai:roncoroni.albertoSummary: We investigate the uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of elliptic equations with a drift posed on a complete, noncompact, Riemannian manifold \(M\) of infinite volume and dimension \(N\geq 2\). Furthermore, in the special case of a model manifold with polynomial volume growth, we show that the conditions on the drift term are sharp.On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problemshttps://zbmath.org/1521.350652023-11-13T18:48:18.785376Z"Audrito, Alessandro"https://zbmath.org/authors/?q=ai:audrito.alessandroSummary: We prove \textit{uniform} parabolic Hölder estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals
\[
{\mathcal{E}}_\varepsilon (W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{\int_{\mathbb{R}_+^{N+1}} y^a \Big(\varepsilon |\partial_t W|^2 + |\nabla W|^2 \Big) \mathrm{d}X + \int_{\mathbb{R}^N \times \{0\}} \Phi (w) \,\mathrm{d}x \bigg\} \,\mathrm{d}t, \qquad \varepsilon \in (0,1)
\]
where \(a \in (-1, 1)\) is a fixed parameter, \(\mathbb{R}_+^{N+1}\) is the upper half-space and \(\mathrm{d}X = \mathrm{d}x \mathrm{d}y\). As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy-Neumann problems arising in combustion theory and fractional diffusion.Weyl's law for the Steklov problem on surfaces with rough boundaryhttps://zbmath.org/1521.351242023-11-13T18:48:18.785376Z"Karpukhin, Mikhail"https://zbmath.org/authors/?q=ai:karpukhin.mikhail-a"Lagacé, Jean"https://zbmath.org/authors/?q=ai:lagace.jean"Polterovich, Iosif"https://zbmath.org/authors/?q=ai:polterovich.iosifSummary: The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as ``slow'' exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result, in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.Fourier coefficients of restrictions of eigenfunctionshttps://zbmath.org/1521.351272023-11-13T18:48:18.785376Z"Wyman, Emmett L."https://zbmath.org/authors/?q=ai:wyman.emmett-l"Xi, Yakun"https://zbmath.org/authors/?q=ai:xi.yakun"Zelditch, Steve"https://zbmath.org/authors/?q=ai:zelditch.steveSummary: Let \(\{e_j\}\) be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold \((M, g)\). Let \(H \subset M\) be a submanifold and \(\{\psi_k\}\) be an orthonormal basis of Laplace eigenfunctions of \(H\) with the induced metric. We obtain joint asymptotics for the Fourier coefficients
\[
\langle\gamma_H e_j, \psi_k \rangle_{L^2 (H)} = \int_H e_j \overline{\psi}_k dV_H
\]
of restrictions \(\gamma_H e_j\) of \(e_j\) to \(H\). In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum \(\left\{ (\mu_k, \lambda_j) \right\}_{j,k-0}^{\infty}\) of the (square roots of the) Laplacian \(\Delta_M\) on \(M\) and the Laplacian \(\Delta_H\) on \(H\) in a family of suitably `thick' regions in \(\mathbb{R}^2\). Thick regions include (1) the truncated cone \(\mu_k / \lambda_j \in [a, b] \subset (0, 1)\) and \(\lambda_j \leqslant \lambda\), and (2) the slowly thickening strip \(|\mu_k - c \lambda_j |\leqslant w (\lambda)\) and \(\lambda_j \leqslant \lambda\), where \(w (\lambda)\) is monotonic and \(1 \ll w (\lambda) \precsim \lambda^{1/2}\). Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.Existence theorems for a generalized Chern-Simons equation on finite graphshttps://zbmath.org/1521.351752023-11-13T18:48:18.785376Z"Gao, Jia"https://zbmath.org/authors/?q=ai:gao.jia"Hou, Songbo"https://zbmath.org/authors/?q=ai:hou.songboSummary: Consider \(G = (V, E)\) as a finite graph, where \(V\) and \(E\) correspond to the vertices and edges, respectively. We study a generalized Chern-Simons equation \(\Delta u = \lambda\mathrm{e}^u(\mathrm{e}^{bu} - 1) + 4\pi\sum_{j = 1}^N \delta_{p_j}\) on \(G\), where \(\lambda\) and \(b\) are positive constants; \(N\) is a positive integer; \(p_1, p_2, \dots, p_N\) are distinct vertices of \(V\); and \(\delta_{p_j}\) is the Dirac delta mass at \(p_j\). We prove that there exists a critical value \(\lambda_c\) such that the equation has a solution if \(\lambda \geq \lambda_c\) and the equation has no solution if \(\lambda < \lambda_c\). We also prove that if \(\lambda > \lambda_c\), the equation has at least two solutions that include a local minimizer for the corresponding functional and a mountain-pass type solution. Our results extend and complete those of \textit{A. Huang} et al. [Commun. Math. Phys. 377, No. 1, 613--621 (2020; Zbl 1447.35338)] and \textit{S. Hou} and \textit{J. Sun} [Calc. Var. Partial Differ. Equ. 61, No. 4, Paper No. 139, 13 p. (2022; Zbl 1491.35238)].
{\copyright 2023 American Institute of Physics}Spike solutions for a fractional elliptic equation in a compact Riemannian manifoldhttps://zbmath.org/1521.351842023-11-13T18:48:18.785376Z"Bendahou, Imene"https://zbmath.org/authors/?q=ai:bendahou.imene"Khemiri, Zied"https://zbmath.org/authors/?q=ai:khemiri.zied"Mahmoudi, Fethi"https://zbmath.org/authors/?q=ai:mahmoudi.fethiSummary: Given an \(n\)-dimensional compact Riemannian manifold \((M,g)\) without boundary, we consider the nonlocal equation
\[\varepsilon^{2s} P_g^s u + u = u^p \quad \hbox{in }\, (M,g),\]
where \(P_g^s\) stands for the fractional Paneitz operator with principal symbol \((-\Delta_g)^s\), \(s \in (0,1)\), \( p \in (1,2_s^*-1)\) with \(2_s^* := \frac{2n}{n-2s} \), \(n>2s\), represents the critical Sobolev exponent and \(\varepsilon > 0\) is a small real parameter. We construct a family of positive solutions \(u_\varepsilon\) that concentrate, as \(\varepsilon \to 0\) goes to zero, near critical points of the mean curvature \(H\) for \(0 <s< \frac{1}{2}\) and near critical points of a reduced function involving the scalar curvature of the manifold~ \(M\) for \( \frac{1}{2} \leq s < 1\).The wave trace and Birkhoff billiardshttps://zbmath.org/1521.370272023-11-13T18:48:18.785376Z"Vig, Amir"https://zbmath.org/authors/?q=ai:vig.amirIn this work a Hadamard-Riesz-type parametrix for the wave propagator in bounded planar domains \(\Omega\) with smooth and strictly convex boundary, is developed.
The derived parametrix is mainly used to produce asymptotic expansions for the distributional wave trace \(\mathrm{Tr} \cos t \sqrt{-\Delta}\) near isolated lengths in the length spectrum, with \(\Delta\) being the Dirichlet Laplacian on a domain \(\Omega\).
Following the work by \textit{S. Marvizi} and \textit{R. Melrose} [J. Differ. Geom. 17, 475--502 (1982; Zbl 0492.53033)], its principal symbol in terms of geometric data associated to the underlying billiard map is computed. This result provides new formulas for the wave invariants.
The article is well written. All the (sometimes technical) proofs are presented step by step. The reference list includes a rich variety of good papers supplementing this research.
Reviewer: Daniel Jaud (München)The Arnold conjecture in \(\mathbb{CP}^n\) and the Conley indexhttps://zbmath.org/1521.370552023-11-13T18:48:18.785376Z"Asselle, Luca"https://zbmath.org/authors/?q=ai:asselle.luca"Izydorek, Marek"https://zbmath.org/authors/?q=ai:izydorek.marek"Starostka, Maciej"https://zbmath.org/authors/?q=ai:starostka.maciejA version of the Arnold conjecture for the complex projective space \(\mathbb{CP} ^n\) endowed with the Fubini-Study form asserts that a Hamiltonian diffeomorphism has at least \(n+1\) fixed points. The result has been proved by \textit{B. Fortune} [Invent. Math. 81, 29--46 (1985; Zbl 0547.58015)]. The authors give an alternative proof of this theorem based on the Hilbert space setting of Conley index theory. This is a part of a broader program of developing methods alternative to Floer theory. The motivation is to improve some known results concerning the degenerate Arnold conjecture on a class of toric manifolds.
Reviewer: Zdzisław Dzedzej (Gdańsk)Cartan connections and integrable vortex equationshttps://zbmath.org/1521.370612023-11-13T18:48:18.785376Z"Ross, Calum"https://zbmath.org/authors/?q=ai:ross.calumThe author investigates the relationship between integrable abelian vortex equations and the geometry of three-dimensional Lie groups. In particular, he shows that an abelian vortex is equivalent to a non-abelian connection. The author introduces and studies the three-dimensional generalization of a vortex, a vortex configuration. He shows that in this case, the vortex configurations are also determined by flat connections, which in this case are pullbacks of the Maurer-Cartan form by means of a bundle map. He also proposes a method for constructing solutions to the massless Dirac equation using vortices.
Reviewer: Viktor Abramov (Tartu)Quantum Bianchi-VII problem, Mathieu functions and arithmetichttps://zbmath.org/1521.370622023-11-13T18:48:18.785376Z"Veselov, A. P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Ye, Y."https://zbmath.org/authors/?q=ai:ye.yiruThis paper is devoted to the geodesic problem on compact threefolds with Riemannian metric of Bianchi-type. This problem is studied in both classical and quantum cases. The authors show that the problem is integrable; they describe explicitly the eigenfunctions of the corresponding Laplace-Beltrami operators in terms of Mathieu functions with parameter depending on the lattice values of some binary quadratic forms. They employ a number-theoretic approach in order to study the level spacing statistics in relation with the Berry-Tabor conjecture and compare the situation when some Bianchi-type metrics are taken into account.
Reviewer: Oleg Karpenkov (Liverpool)Cartan motion group and orbital integralshttps://zbmath.org/1521.430032023-11-13T18:48:18.785376Z"Song, Yanli"https://zbmath.org/authors/?q=ai:song.yanli.1|song.yanli"Tang, Xiang"https://zbmath.org/authors/?q=ai:tang.xiang|tang.xiang.1Summary: In this short note, we study the variation of orbital integrals, as traces on the group algebra $G$, under the deformation groupoid. We show that orbital integrals are continuous under the deformation. And we prove that the pairing between orbital integrals and \(K\)-theory element of \(C^*_r(G)\) stays constant with respect to the deformation for regular group elements, but vary at singular elements.
For the entire collection see [Zbl 1507.19001].Cyclic cohomology and the extended Heisenberg calculus of Epstein and Melrosehttps://zbmath.org/1521.460352023-11-13T18:48:18.785376Z"Gorokhovsky, Alexander"https://zbmath.org/authors/?q=ai:gorokhovsky.alexander"van Erp, Erik"https://zbmath.org/authors/?q=ai:van-erp.erikSummary: In this paper we present a formula for the index of a pseudodifferential operator with invertible principal symbol in the \textit{extended} Heisenberg calculus of Epstein and Melrose. Our results build on the work we did in a previous paper
[\textit{A.~Gorokhovsky} and \textit{E.~van Erp}, Adv. Math. 399, Article ID 108229, 62~p. (2022; Zbl 1489.58008)], where we restricted attention to the Heisenberg calculus proper.
For the entire collection see [Zbl 1507.19001].Cyclic cocycles and quantized pairings in materials sciencehttps://zbmath.org/1521.460372023-11-13T18:48:18.785376Z"Prodan, Emil"https://zbmath.org/authors/?q=ai:prodan.emilSummary: The pairings between the cyclic cohomologies and the K-theories of separable \(C^*\)-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify how some of these invariants survive throughout the full Sobolev domains of the cocycles. These interesting phenomena, which can be explained by index theorems derived from Alain Connes' quantized calculus, are now well understood in the independent electron picture. Here, we review recent developments addressing the dynamics of correlated many-fermions systems, obtained in collaboration with Bram Mesland. They supply a complete characterization of an algebra of relevant derivations over the \(C^*\)-algebra of canonical anti-commutation relations indexed by a generic discrete Delone lattice. It is argued here that these results already supply the means to generate interesting and relevant states over this algebra of derivations and to identify the cyclic cocycles corresponding to the transport coefficients of the many-fermion systems. The existing index theorems for the pairings of these cocycles, in the restrictive single fermion setting, are reviewed and updated with an emphasis on pushing the analysis on Sobolev domains. An assessment of possible generalizations to the many-body setting is given.
For the entire collection see [Zbl 1507.19001].On the noncommutative mapping torus and related structureshttps://zbmath.org/1521.460392023-11-13T18:48:18.785376Z"Rezaei, A. A."https://zbmath.org/authors/?q=ai:rezaei.ali-asghar|rezaei.aliabad-aliSummary: In this paper the noncommutative mapping torus and some of its related structures are investigated. Some generalizations are also introduced. One of these generalizations is noncommutative mapping torus telescope which is defined for a sequence of \(C^*\)-morphisms and the other is double mapping which applied for a couple of \(C^*\)-morphisms between two \(C^*\)-algebras. Finally, we generalize the double mapping for a couple of parallel sequences of \(C^*\)-morphisms in order to introduce the double mapping telescope.\(p\)-Schatten commutators of projectionshttps://zbmath.org/1521.470392023-11-13T18:48:18.785376Z"Andruchow, Esteban"https://zbmath.org/authors/?q=ai:andruchow.esteban"Di Iorio y Lucero, María Eugenia"https://zbmath.org/authors/?q=ai:di-iorio-y-lucero.maria-eugeniaSummary: Let \(\mathcal{H}=\mathcal{H}_+\oplus\mathcal{H}_-\) be a fixed orthogonal decomposition of the complex separable Hilbert space \(\mathcal{H}\) in two infinite-dimensional subspaces. We study the geometry of the set \(\mathcal{P}^p\) of selfadjoint projections in the Banach algebra
\[
\mathcal{A}^p=\{A\in\mathcal{B}(\mathcal{H}): [A,E_+]\in\mathcal{B}_p(\mathcal{H})\},
\]
where \(E_+\) is the projection onto \(\mathcal{H}_+\) and \(\mathcal{B}_p(\mathcal{H})\) is the Schatten ideal of \(p\)-summable operators \((1\leq p<\infty)\). The norm in \(\mathcal{A}^p\) is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space \(\mathcal{P}^p\) is shown to be a differentiable \(C^\infty\) submanifold of \(\mathcal{A}^p\), and a homogeneous space of the group of unitary operators in \(\mathcal{A}^p\). The connected components of \(\mathcal{P}^p\) are characterized, by means of a partition of \(\mathcal{P}^p\) in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.Optimal control results for Sobolev-type fractional stochastic Volterra-Fredholm integrodifferential systems of order \(\vartheta \in (1, 2)\) via sectorial operatorshttps://zbmath.org/1521.490052023-11-13T18:48:18.785376Z"Johnson, M."https://zbmath.org/authors/?q=ai:johnson.michael-james|johnson.mark-w|johnson.murugesan|johnson.mark-r|johnson.matthew-leander|johnson.michael-d|johnson.mija-salomi|johnson.mark-a|johnson.mikala-c|johnson.malvin-g-jun|johnson.megan|johnson.mark-j|johnson.michael-joseph|johnson.marvin-m|johnson.margaret-c|johnson.marjory-j|johnson.marina|johnson.micah-k|johnson.mathew-a|johnson.michael-c-r|johnson.matthew-a|johnson.michael-a|johnson.merek|johnson.mikkel-b|johnson.matthew-c|johnson.michael-s|johnson.marcus|johnson.miki|johnson.marianne|johnson.margaret-a|johnson.michael-t|johnson.michael-p|johnson.m-eric|johnson.m-v|johnson.madeline|johnson.michele-l|johnson.m-f-g|johnson.marilynn|johnson.matthew-scott|johnson.matthew-p|johnson.maribeth|johnson.mac|johnson.matthew-r|johnson.mark-e|johnson.millard-w-jun|johnson.mark-p|johnson.matthew|johnson.michael-b|johnson.martin.1|johnson.mary-holland|johnson.michael-s-j|johnson.mary-a|johnson.matthew-james|johnson.marla|johnson.mark-scott"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.vaidehi|vijayakumar.velusamyAuthors' abstract: The objective of this paper is to investigate the optimal control results for Sobolev-type fractional stochastic Volterra-Fredholm integrodifferential systems of order \(\vartheta \in (1, 2)\) with sectorial operators in Hilbert spaces. Initially, we prove the existence of mild solutions by using fractional calculus, stochastic analysis theory, and the Schauder's fixed point theorem. Next, we demonstrate the existence of optimal control pairs for the given system. Finally, an example is included to show the applications of the developed theory.
Reviewer: Ahmed M. A. El-Sayed (Alexandria)Optimality conditions for approximate solutions of nonsmooth semi-infinite vector optimization problemshttps://zbmath.org/1521.490162023-11-13T18:48:18.785376Z"Jiao, Liguo"https://zbmath.org/authors/?q=ai:jiao.liguo"Kim, Do Sang"https://zbmath.org/authors/?q=ai:kim.do-sangUsing Ekeland Variational Principle and Mordukhovich generalized differentiation, first the paper provides fuzzy necessary optimality condition for weak \(\epsilon\)-Pareto solutions to a class of nonsmooth semi-infinite vector optimization problem. Then the exact necessary optimality condition for weak quasi \(\epsilon\)-Pareto solutions is established by considering an associated optimization problem with an limiting constraint qualification. A simple example shows that the limiting constraint qualification cannot be removed in the latter result.
For the entire collection see [Zbl 1508.49001].
Reviewer: Ba Khiet Le (Ho Chi Minh City)Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifoldshttps://zbmath.org/1521.490352023-11-13T18:48:18.785376Z"Wang, Yu-Zhao"https://zbmath.org/authors/?q=ai:wang.yuzhao"Li, Sheng-Jie"https://zbmath.org/authors/?q=ai:li.shengjie"Zhang, Xinxin"https://zbmath.org/authors/?q=ai:zhang.xinxinSummary: In this paper, we prove the generalized displacement convexity for nonlinear mobility continuity equation with \(p\)-Laplacian on Wasserstein space over Riemannian manifolds under the generalized McCann condition GMC\((m, n)\). Moreover, we obtain some variational formulae along the Langevin deformation of flows on the generalized Wasserstein space, which is the interpolation between the gradient flow and the geodesic flow. We also establish the connection between the displacement convexity of entropy functionals and the concavity of \(p\)-Rényi entropy powers. As an application, we derive the NIW formula which indicates the relationship between the \(p\)-Rényi entropy powers \({\mathcal{N}}_b\), the Fisher information \({\mathcal{I}}_b\) and the \({\mathcal{W}}\)-entropy.Geometric theory of Weyl structureshttps://zbmath.org/1521.530142023-11-13T18:48:18.785376Z"Čap, Andreas"https://zbmath.org/authors/?q=ai:cap.andreas"Mettler, Thomas"https://zbmath.org/authors/?q=ai:mettler.thomasGiven a parabolic geometry on a smooth manifold \(M\), the authors study a natural affine bundle \(A\to M\) whose smooth sections can be identified with Weyl structures.
In particular, the paper considers torsion-free AHS structures:
Let \(g\) be a semisimple Lie algebra endowed with a \(|1|\)-grading, i.e., with a decomposition
\[ g= g_{-1}\oplus g_0\oplus g_1 \]
into a direct sum of linear subspaces such that:
(1) \([ g_i, g_j]\subset g_{i+j}\) setting \( g_\ell=\{0\}\) if \(|\ell|>1\);
(2) No simple ideal of \( g\) is contained in \( g_0\).
Then \( \widetilde p= g_0+ g_1\) is a Lie subalgebra of \( g\). Let \(G\) be a Lie group with Lie algebra \(g\). The normalizer of \(\widetilde p\) in \(G\) has Lie algebra \({p}\). Choosing a closed subgroup \(P\subset G\) lying between this normalizer and its connected component of the identity, a parabolic geometry of type \((G,P)\) is a Cartan geometry \((p:\mathcal{G}\to M,\omega)\) of type \((G,P)\).
Definition. For the parabolic geometry \((p:\mathcal G\to M,\omega)\) the associated bundle of Weyl structures is
\[ \pi:A:=\mathcal{G}\times_P(P/G_0)\to M, \]
where \(G_0\subset P\) is the closed subgroup of elements whose adjoint action preserves the grading of \(g\).
Weyl structures for parabolic geometries were originally introduced by \textit{A. Čap} and \textit{J. Slovák} [Math. Scand. 93, No. 1, 53--90 (2003; Zbl 1076.53029)] as \(G_0\)-equivariant sections of the natural projection \(q:\mathcal G\to\mathcal G_0:=\mathcal G/P_+\). Such sections correspond to smooth sections of \(\pi:A\to M\) (see \S2.2), such that the space of sections of \(A\to M\) can be naturally identified with the space of Weyl structures for the geometry \((p:\mathcal G\to M,\omega)\).
The initial parabolic geometry is shown to define a reductive Cartan geometry on \(A\), inducing an almost bi-Lagrangian structure \((\Omega,L^+,L^-)\) on \(A\) and a compatible linear connection on \(TA=L^-\oplus L^+\) (see \S2.3 and \S3.1). The skew symmetric bilinear bundle map \(\Omega\) defines an almost symplectic structure. One of the main theorems concerns the case when it is symplectic.
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a parabolic geometry of type \((G,P)\) and \(\pi:A\to M\) its associated bundle of Weyl structures. Then the natural 2-form \(\Omega\in\Omega^2(A)\) is closed if and only if \((G,P)\) corresponds to a \(|1|\)-grading and the Cartan geometry \((p:\mathcal{G}\to M,\omega)\) is torsion-free.
The canonical almost bi-Lagrangian structure on the bundle \(A\to M\) also induces a canonical nondegenerate bilinear form \(h\).
Theorem. For any torsion-free AHS structure, the pseudo-Riemannian metric \(h\) induced by the canonical almost bi-Lagrangian structure on \(A\to M\) is an Einstein metric with nonzero scalar curvature.
A Weyl structure of a torsion-free AHS structure is Lagrangian if \(s:M\to(A,\Omega)\) is a Lagrangian submanifold, and nondegenerate if \(s:M\to(A,h)\) is a nondegenerate submanifold. Lagrangian Weyl structures that lead to totally geodesic submanifolds \(s(M)\subset A\) are characterized as follows (the Rho-tensor is introduced in \S2.2):
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a torsion-free AHS structure and \(\pi:A\to M\) its bundle of Weyl structures. Let \(s:M\to A\) be a smooth section corresponding to a Lagrangian Weyl structure, let \(\nabla^s\) and \(\mathsf{P}^s\) denote the corresponding Weyl connection and Rho-tensor, respectively. Then the following conditions are equivalent:
(1) The submanifold \(s(M)\subset A\) is totally geodesic for the canonical connection \(D\) (\S2.3);
(2) The submanifold \(s(M)\subset A\) is totally geodesic for the Levi-Civita connection of \(h\);
(3) \(\nabla^s\mathsf P^s = 0\).
This provides a connection to Einstein metrics and reductions of projective holonomy (\S3.4).
Moreover, given nondegeneracy, a universal formula for the second fundamental form of this image is obtained. The second fundamental forms of \(s:M\to(A,h)\) with respect to \(D\) and the Levi-Civita connection of \(h\) are, respectively, denoted by \(\mathrm{I\!I}^s_D\) and \(\mathrm{I\!I}^s_h\).
Theorem. Let \((p:\mathcal{G}\to M,\omega)\) be a torsion-free AHS structure and \(\pi:A\to M\) its bundle of Weyl structures. Let \(s:M\to A\) be a nondegenerate, Lagrangian Weyl structure with Weyl connection \(\nabla^s\) and Rho-tensor \(\mathsf{P}\). Then \[ \mathrm{I\!I}^s_D=-\frac12\mathsf{P}^{ka}\nabla^s_i\mathsf{P}_{ja} \] and \[ \mathrm{I\!I}^s_h = -\frac12\mathsf{P}^{ka}\,(\nabla^s_i\mathsf{P}_{ja}+\nabla^s_j\mathsf{P}_{ia}-\nabla^s_a\mathsf{P}_{ij}). \]
In \S4, relations between the geometry on \(A\) and nonlinear invariant partial differential equations are discussed. For locally flat projective structures, interrelations to solutions of a projectively invariant Monge-Ampère equation are obtained. Theorem 4.4 relates Calabi's equation to an equation for a minimal Lagragian Weyl structure. As a consequence, the following statement is obtained:
Corollary. Let \((M, [\nabla])\) be a closed oriented locally flat projective manifold. Then \([\nabla]\) is properly convex if and only if it arises from a minimal Lagrangian Weyl structure whose Rho-tensor is positive definite.
Reviewer: Andreas Vollmer (Hamburg)Holonomy pseudogroups of manifolds over Weil algebrashttps://zbmath.org/1521.530162023-11-13T18:48:18.785376Z"Shurygin, V. V."https://zbmath.org/authors/?q=ai:shurygin.vadim-v-jun"Zubkova, S. K."https://zbmath.org/authors/?q=ai:zubkova.svetlana-kSummary: The notion of the holonomy pseudogroup on a total immersed transversal is extended to the case of complete foliated smooth manifolds over a Weil algebra \({\mathbf{A}}\) modelled on \({\mathbf{A}} \)-modules of the type \({\mathbf{A}}^n\oplus{\mathbf{B}}^m \), where \({\mathbf{B}}\) is a quotient algebra of \({\mathbf{A}} \). It is proved that the holonomy pseudogroup determines a complete \({\mathbf{A}} \)-smooth manifold up to \({\mathbf{A}} \)-diffeomorphism, and examples of application of holonomy pseudogroups are presented.De Rham cohomology and semi-slant submanifolds in metallic Riemannian manifoldshttps://zbmath.org/1521.530172023-11-13T18:48:18.785376Z"Gök, Mustafa"https://zbmath.org/authors/?q=ai:gok.mustafaSummary: In this paper, we deal with the de Rham cohomology of semi-slant submanifolds in metallic Riemannian manifolds. Some necessary conditions for a semi-slant submanifold of metallic Riemannian manifolds are given to define a well-defined canonical de Rham cohomology class. Also, the non-triviality of such a cohomology class is discussed. Finally, an example is constructed to illustrate the main idea of the paper.Variational problems concerning sub-Finsler metrics in Carnot groupshttps://zbmath.org/1521.530222023-11-13T18:48:18.785376Z"Essebei, Fares"https://zbmath.org/authors/?q=ai:essebei.fares"Pasqualetto, Enrico"https://zbmath.org/authors/?q=ai:pasqualetto.enricoLet \({G}\) be a Carnot group equipped with a left-invariant sub-Riemannian metric \(\langle \cdot , \cdot \rangle\) on the horizontal tangent bundle \(H {G}\) induced by the first stratum of its Lie algebra, and denote by \(d_{cc}\) the induced Carnot-Carathéodory distance on \({G}\). Let \(\Omega \subset {G}\) be open. The main objects of interest in this paper are distance functions \(d\) on \(\Omega\) which are geodesic and bi-Lipschitz equivalent to \(d_{cc}\).
Section 4 is concerned with convergence properties for such distances. Given a distance \(d\) on \(\Omega\) satisfying the above conditions, consider the following functionals:
(1) For a Lipschitz path \(\gamma \in \mathrm{Lip}([0,1], \Omega)\), let \(L_d(\gamma)\) be the length of \(\gamma\) with respect to \(d\);
(2) For a positive Borel measure \(\mu\) on \(\Omega \times \Omega\), let \(J_d(\mu) = \int d(x,y) \mu(dx, dy)\). When \(\mu\) is a probability measure, \(J_d(\mu)\) can be interpreted as a coupling of two probability measures on \(\Omega\), and then it represents the transportation cost of this coupling.
Now let \(d_n\) be a sequence of such distances on \(\Omega\), with \(d\) another such distance, and suppose that their bi-Lipschitz constants are uniformly bounded by some \(\alpha \ge 1\). (In the paper, the latter condition is easy to overlook, as it is implicit in the definition of the class \(\mathcal{D}_{cc}(\Omega)\).) Define the functionals \(L_n, J_n\) accordingly from \(d_n\). In Theorem 4.4, the authors show that the following conditions are equivalent:
(1) \(d_n \to d\) uniformly on compact subsets of \(\Omega \times \Omega\);
(2) The sequence \(L_n\) is \(\Gamma\)-convergent to \(L\). This means that if \(\gamma_n \to \gamma\) uniformly on \([0,1]\), then \(L(\gamma) \le \liminf L_n(\gamma_n)\); and for each path \(\gamma\) there exists a sequence \(\gamma_n\) with \(\gamma_n \to \gamma\) uniformly and \(\limsup L_n(\gamma_n) \le L(\gamma)\);
(3) The sequence \(J_n\) is \(\Gamma\)-convergent to \(J\) in a similar sense. Here we ask for the sequences \(\mu_n\) to converge to \(\mu\) in the weak-* topology.
Moreover, if \(\Omega\) is bounded, we can strengthen (3) to \(J_n(\mu_n) \to J(\mu)\) whenever \(\mu_n \to \mu\) in the weak-* topology.
The other main topic of the paper is the relationship between geodesic distance functions on \(\Omega\) and sub-Finsler convex metrics. Here, a sub-Finsler convex metric on \(\Omega\) is a measurable function \(\varphi : H \Omega \to [0, \infty)\) on the horizontal distribution, which is a norm on each fiber \(H_x\). As with the distance functions, the authors restrict attention to those sub-Finsler convex metrics which are bi-Lipschitz equivalent to the sub-Riemannian metric, so that for some \(\alpha \ge 1\) we have \(\frac{1}{\alpha} \|v\| \le \varphi(v) \le \alpha \|v\|\), where \(\|v\|^2 = \langle v, v \rangle\). Such a \(\varphi\) induces a dual norm \(\varphi^\star\) on the dual bundle \(H^* \Omega\), which may be identified again with \(H \Omega\) via the sub-Riemannian metric.
A distance function \(d\) may be differentiated along horizontal directions to produce a sub-Finsler convex metric \(\varphi_d\) (Theorem 3.9). Conversely, given a sub-Finsler convex metric \(\varphi\), there are two natural ways to produce a distance function. On the one hand, we have the intrinsic distance defined via lengths of paths, as
\[
d_\varphi(x,y) = \inf_\gamma \int_0^1 \varphi(\dot{\gamma}(t))\,dt
\]
with the infimum taken as usual over horizontal paths joining \(x\) and \(y\). On the other hand, we can define
\[
\delta_\varphi(x,y) = \sup\{ |f(x) - f(y)| : \varphi(\nabla f) \le 1\}
\]
where \(\nabla\) denotes the horizontal gradient with respect to the reference sub-Riemannian metric \(\langle \cdot, \cdot \rangle\). This construction is in some sense dual to the intrinsic distance, as will be seen.
The results of the paper include:
(1) If \(d\) is a geodesic distance that is bi-Lipschitz equivalent to \(d_{cc}\), it is the intrinsic distance of its metric derivative: \(d = d_{\varphi_d}\) (Theorem 3.5);
(2) If we begin with a sub-Finsler convex metric \(\psi\) and take the metric derivative \(\varphi_{d_\psi}\) of its intrinsic distance, we do not in general recover \(\psi\). However, we do have \(\varphi_{d_\psi} \le \psi\) on almost every fiber \(H_x\), and on every fiber if \(\psi\) is upper semicontinuous. We obtain the opposite inequality if \(\psi\) is lower semicontinuous (Theorem 5.9);
(3) For any sub-Finsler convex metric \(\varphi\), we have \(\delta_\varphi \le d_{\varphi^\star}\), with equality if \(\varphi\) is lower semicontinuous or upper semicontinuous (Theorems 5.11 and 5.12);
(4) If \(\varphi\) is a sub-Finsler convex metric which is upper semicontinuous, then \(\varphi(\nabla f)\) equals the local Lipschitz constant of \(f\) with respect to \(\delta_\varphi\) almost everywhere.
Reviewer: Nathaniel Eldredge (Greeley)Nowhere differentiable intrinsic Lipschitz graphshttps://zbmath.org/1521.530242023-11-13T18:48:18.785376Z"Julia, Antoine"https://zbmath.org/authors/?q=ai:julia.antoine"Nicolussi Golo, Sebastiano"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastiano"Vittone, Davide"https://zbmath.org/authors/?q=ai:vittone.davideThe paper gives examples of Lipschitz graphs that are nowhere intrinsically differentiable.
The main theorem reads:
Theorem. Let \( G\) be a Carnot group with stratification \(\bigoplus_{j=1}^{s} V_j\). Let \({W}{ V}\) be a splitting of \(G\) such that \({W}V_2\not\subset [{ W},{W}]\) and there exists \(v_0 \in {V}\cap V_1\) such that \(v_0\not= 0\) and \([v_0,{W}]=0\).
Then there is an intrinsic Lipschitz function \(\phi: {W} \rightarrow { V}\) that is nowhere intrinsically differentiable.
Moreover, \(\phi\) can be constructed in such a way that, for every \(p\in \Gamma_{\phi}\), the following properties hold:
(a) There exist infinitely many different blow-ups of \(\Gamma_{\phi}\) at \(p\);
(b) No blow-up of \(\Gamma_{\phi}\) at \(p\) is a homogeneous subgroup.
Reviewer: Sergei V. Rogosin (Minsk)A prescribed scalar and boundary mean curvature problem and the Yamabe classification on asymptotically Euclidean manifolds with inner boundaryhttps://zbmath.org/1521.530272023-11-13T18:48:18.785376Z"Sicca, Vladmir"https://zbmath.org/authors/?q=ai:sicca.vladmir"Tsogtgerel, Gantumur"https://zbmath.org/authors/?q=ai:tsogtgerel.gantumurSummary: We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary and sufficient condition in terms of a conformal invariant of the zero sets of the target curvatures for the existence of solutions to the problem and use this result to establish the Yamabe classification of metrics in those manifolds with respect to the solvability of the prescribed curvature problem.Some characterizations of spheres by conformal vector fieldshttps://zbmath.org/1521.530282023-11-13T18:48:18.785376Z"Ye, Jian"https://zbmath.org/authors/?q=ai:ye.jianSummary: In this paper, we consider conformal characterizations of standard sphere in terms of conformal vector fields on closed Riemannian manifolds. We firstly prove that each closed Riemannian manifold with Ricci curvature being non-negative in certain direction and constant scalar curvature is isometric to standard sphere if and only if it admits a non-trivial closed conformal vector field. In the case of non-constant scalar curvature, we show that each closed Riemannian manifold of dimension two with positive Gauss curvature carrying a non-trivial closed conformal vector field is conformal to a round sphere and we generalize the result to high dimensions in two directions.Poincaré type inequality for hypersurfaces and rigidity resultshttps://zbmath.org/1521.530292023-11-13T18:48:18.785376Z"Alencar, Hilário"https://zbmath.org/authors/?q=ai:alencar.hilario"Batista, Márcio"https://zbmath.org/authors/?q=ai:batista.marcio"Silva Neto, Gregório"https://zbmath.org/authors/?q=ai:silva-neto.gregorioThe authors prove a general Poincaré type inequality on Riemannian manifolds whose sectional curvature is suitably bounded. They then apply this inequality, combined with some other mild assumptions, to prove the following:
(1) Some isoperimetric inequalities for domains of hypersurfaces;
(2) Minimal hypersurfaces of space forms (satisfying suitable decay properties) are foliated by totally geodesic submanifolds;
(3) Hypersurfaces with a determined constant scalar curvature in Einstein manifolds are totally geodesic;
(4) The hyperplane is rigid, in the sense that is the only homothetic self-similar solution to a class of fully nonlinear curvature flows.
Reviewer: Giorgio Saracco (Firenze)Conformal Dirac-Einstein equations on manifolds with boundaryhttps://zbmath.org/1521.530302023-11-13T18:48:18.785376Z"Borrelli, William"https://zbmath.org/authors/?q=ai:borrelli.william"Maalaoui, Ali"https://zbmath.org/authors/?q=ai:maalaoui.ali"Martino, Vittorio"https://zbmath.org/authors/?q=ai:martino.vittorioLet \(M\) be a compact oriented 3-dimensional manifold with nonempty boundary \(\partial M\). Let the spin structure be fixed on \(M\) and \(\partial M\) carry the one canonically induced by the outward normal along \(\partial M\), whatever the Riemannian metric is. A new functional of Gibbons-Hawking-York type is introduced and its critical points within a conformal class are investigated. This functional \(\mathcal{E}\) is defined by
\[
\mathcal{E}(g,\psi):=\int_M R_g\,dv_g+\frac{1}{2}\int_{\partial M}h_g\,d\sigma_g+\int_M\left(\langle D_g\psi,\psi\rangle-\langle\psi,\psi\rangle\right) \,dv_g,
\]
where \(g\) is a Riemannian metric on \(M\) with scalar curvature \(R_g\) and boundary mean curvature \(h_g\), and \(\psi\) is a spinor field on \((M,g)\). As usual, \(D_g\) denotes the Dirac operator associated to the metric \(g\) and the spin structure on \(M\).
The functional \(\mathcal{E}\) extends both the Gibbons-Hawking-York functional, which generalizes the Einstein-Hilbert functional on manifolds with boundary, and the Einstein-Dirac functional on closed \(3\)-manifolds.
The functional \(\mathcal{E}\) is restricted to a conformal class with fixed boundary volume, in which case the metric \(g\) can be expressed in terms of a real function \(u\) and a supplementary summand involving a real nonnegative parameter \(b\). The critical points of that new functional \((u,\psi)\mapsto E^b(u,\psi)\) are first characterized assuming chiral bag boundary conditions for the Dirac operator. The Euler-Lagrange equations involve a linearising operator \(L_g\) which, under the above boundary conditions, turns out to be elliptic and self-adjoint and thus to have a discrete spectrum consisting of real eigenvalues of finite multiplicities. Assuming the Yamabe invariant of \((M,\partial M)\) w.r.t. the chosen conformal class to be positive, which is equivalent to the first eigenvalue of \(L_g\) being positive, the authors consider sequences converging to critical points of \(E^b\) and the corresponding expected blow-up phenomena. The first main result (Theorem 1.1) gives an asymptotic expansion for a Palais-Smale sequence in both cases where the blow-up occurs in the interior and on the boundary of \(M\). Along the way, a rigidity result is established in the case where \(b=0\) and \(u\geq0\) (Theorem 1.4). The second main result (Theorem 1.6) states an Aubin-type inequality for the Yamabe invariant of \((M,\partial M)\) in the case \(b=0\).
The article is structured as follows. After the presentation of the main results in Section 1, preliminaries about the operators and their conformal covariance are discussed in Section 2. A regularity result (Theorem 3.1) is then established in Section 3. Theorem 1.1 is proved in Section 4, while the rigidity result for \(b=0\) is dealt with in Section 5. Section 6 contains the proof of the Aubin-type inequality.
Reviewer: Nicolas Ginoux (Metz)The spinor and tensor fields with higher spin on spaces of constant curvaturehttps://zbmath.org/1521.530392023-11-13T18:48:18.785376Z"Homma, Yasushi"https://zbmath.org/authors/?q=ai:homma.yasushi"Tomihisa, Takuma"https://zbmath.org/authors/?q=ai:tomihisa.takumaThe paper under review considers the standard first-order differential operators \(D_j\) on spinor fields with spin \(j+1/2\) over Riemannian spin manifolds. The case \(j=0\) corresponds to the classical Dirac operator, and the Rarita-Schwinger operator occurs for \(j=1\).
After several observations in the general case in Section 2, the authors focus on spaces of constant sectional curvature. They compute for the round sphere the spectra of \(D_j\) and some other related operators, including the standard Laplacian acting on the smooth sections of the mentioned fiber bundle.
The cases of trace-free symmetric tensor fields and spinor fields coupled with differential forms are also considered.
Reviewer: Emilio A. Lauret (Bahía Blanca)A variational characterization of calibrated submanifoldshttps://zbmath.org/1521.530422023-11-13T18:48:18.785376Z"Cheng, Da Rong"https://zbmath.org/authors/?q=ai:cheng.da-rong"Karigiannis, Spiro"https://zbmath.org/authors/?q=ai:karigiannis.spiro"Madnick, Jesse"https://zbmath.org/authors/?q=ai:madnick.jesseThe typical setting in calibrated geometries consists of a manifold and a calibration compatible with (or inducing) a Riemannian metric on the manifold, and then possibly the search for the calibrated submanifolds, which are in particular of minimal volume due to the properties of the calibration being both of comass 1 and a closed differential form.
In this article, the authors begin with a Riemannian manifold \((\overline{M},\overline{g})\), consider any given closed embedded \(k\)-submanifold \(M\subset \overline{M}\), and study the volume functional \(\overline{g}\longmapsto\mathcal{V}(\overline{g})= \mathrm{vol}(M,g)= \int_M\mathrm{vol}_{(M,g)}\) where \(g=\overline{g}_{|M}\).
The functional clearly has no critical points on the space of all metrics on \(\overline{M}\), so a novel kind of calibration theory is designed in this paper. The problem was also studied in [\textit{C. Arezzo} and \textit{J. Sun}, J. Reine Angew. Math. 709, 171--200 (2015; Zbl 1331.53105); \textit{J. Sun}, J. Math. Anal. Appl. 434, 1474--1488 (2016; Zbl 1328.53096); \textit{Q. Tan}, Ann. Global Anal. Geom. 53, 217--231 (2018; Zbl 1468.53021)].
The manifold is now assumed to be endowed with a set of \(k\)-forms \(\mu\), not necessarily closed. Then the variational calculus of \(\mathcal V\) is restricted to those metrics which live in a subclass \(\mathcal G\) determined by \(\mu\), where a fortiori it is required that \(\overline{g}\) keeps \(\mu\) with comass 1.
A general and main assertion is the so-called ``meta-theorem''. Roughly, condensed in Theorems A and B in the paper, it says that the metric \(\overline{g}\) on \(\overline{M}\) is a critical point of \(\mathcal V\) if and only if \(M\) is calibrated by \(\mu\), this is, \(\mu_{|M}=\mathrm{vol}_{(M,g)}\). The result is established rigorously and developed \textit{only} in four specific contexts or geometries:
(1) The first case is that of a Hermitian structure \((\overline{M},\overline{g},J,\omega)\). Here, the family \(\mathcal G\) is given as the metrics on \(\overline{M}\) of the form \(\overline{g}_t(X,Y) = \omega_t(X,JY)\), where \(\omega_t =\omega + \mathrm{d}\alpha_t^{(1,1)}\) and \(\alpha_t\) is a one-parameter family of compactly supported smooth 1-forms on \(\overline{M}\) with \(\alpha_{0}=0\), for \(|t|\) sufficiently small so as to assure positive definiteness. All metrics define Hermitian structures (eventually with torsion) compatible with the same \(J\). The \(2k\)-forms \(\mu\) are now the typical forms \(\frac{1}{k!}\omega_t^k\) and the fixed oriented closed \(2k\)-submanifold \(M\) is a complex submanifold of \((\overline{M},J)\).
(2) The second case is a similar variation of a metric induced by a \(\mathrm{G}_2\)-structure \(\varphi\) on a given 7-dimensional manifold \((\overline{M},\varphi)\). As it is well known, the structure 3-form \(\varphi\) induces a metric \(\overline{g}\). Hence the authors may consider small deformations via \(\varphi_t=\varphi+\mathrm{d}\beta_t\) with \(\beta_{0}=0\). The submanifold \(M\) is a 3-manifold, and if calibrated it is associative. Equivalently, \(\overline{g}\) is critical for the functional \(\mathcal V\) in the directions of \(\mathrm{d}\dot{\beta}={\frac{d}{dt}}_{\mid_{t=0}}{\varphi}_t\). Indeed the main theorem holds true for the starting \(\mathrm{G}_2\)-structure.
(3) The third case is that of the calibration \(\star\varphi\) and the corresponding calibrated and coassociative submanifolds.
(4) The fourth case is for \(\mathrm{Spin}(7)\) structures on 8-manifolds and the Cayley submanifolds. Here the family of variations of the metric has to be carefully chosen and theorems A and B reveal marked differences from the first three cases.
A difficult task in all four contexts seems to be proving Theorem B, i.e., to see that \(M\) is calibrated by \(\mu\) if \(\overline{g}\) is a critical point of \(\mathcal V\), and moreover to see that \(M\) is of the required type, respectively, complex, associative, coassociative or Cayley.
The last section is devoted to observations and generalisations of the variational characterization of submanifolds. Firstly, the more abstract proof of Theorem A in all cases, i.e., to see that \(\overline{g}\) is a critical point of \(\mathcal V\) if \(M\) is calibrated by \(\mu\). Secondly, considering the possible role of the Berger-Ebin theorem in the study of such variations. Thirdly, the framework of variations of the metric on \(\overline{M}\) induced by the \(k\)-forms \(f_t^*\mu\) defined by the flow of diffeomorphisms \(f_t\) of a given vector field on \(\overline{M}\). Fourthly, an application of the theory to the variational characterization of the so-called Smith maps, providing further a kind of Theorems A and B about the equivalence between such maps and critical points of the \(k\)-energy.
Reviewer: Rui Albuquerque (Lisboa)Harmonic metrics, harmonic tensors and their applicationshttps://zbmath.org/1521.530452023-11-13T18:48:18.785376Z"Chen, Bang-Yen"https://zbmath.org/authors/?q=ai:chen.bang-yenIn this survey paper, the author summarizes the main results from the mathematical literature concerning the harmonicity of metrics and tensors. The research in this field was initiated in [the author and \textit{T. Nagano}, J. Math. Soc. Japan 36, 295--313 (1984; Zbl 0543.58015)], where a harmonic metric was defined as a metric \(g'\) on a Riemannian manifold \((M, g)\), such that the identity map \(\mathrm{id}_M : (M, g) \rightarrow (M,g')\) is harmonic. For the harmonicity of a metric \(g'\) on a Riemannian manifold \((M,g)\) the author obtains firstly a characterization in terms of the Christoffel symbols with respect to a local coordinate system of \(M\), namely
\[
g^{ji}\left({\Gamma'}_{ji}^k-\Gamma_{ji}^k\right)=0,
\]
and secondly a characterization which involves the function \(f := \mathrm{ tr}_gg'\) and a vector field \(\omega\) with components \(\omega_j=\nabla^kg'_{kj}\) satisfying the relation
\[\nabla_jf=2\omega_j. \]
A symmetric tensor field of type \((0,2)\) on a Riemannian manifold \((M,g)\) satisfying such a condition is called a harmonic tensor field. The author shows that the space of all harmonic symmetric tensors of type \((0,2)\) on a Riemannian manifold \((M,g)\) of dimension \(n\) is isomorphic to the space of all conservative symmetric tensors of type \((0,2)\) on \((M,g)\) if \(n\geq 3\), and it decomposes into the direct sum
\[ \{\lambda g,\ \lambda \text{ smooth function on }M\}\oplus\{T\in\ker(\delta): \mathrm{tr} T=0\} \]
if \(n=2\). Then the author proves that for any Riemannian manifold \((M,g)\) of dimension \(\geq 2\) the Ricci tensor is harmonic with respect to \(g\) and the conservativity of the Ricci tensor is equivalent to the property of \((M,g)\) having constant scalar curvature. Moreover, if \((M,g)\) is an orientable 2-dimensional Riemannian manifold, then the space \(\{T \in \ker(\delta) : \mathrm{tr}_g T = 0\}\) is linearly isomorphic to the space of holomorphic quadratic differentials on \((M, g)\) endowed with the natural complex structure. If \(M\) is diffeomorphic either to a 2-sphere or to a real projective plane, then all harmonic metrics on \((M, g)\) are conformal changes of the metric \(g\).
The author shows that the nullity of the identity map of a Riemannian manifold is equal to the dimension of the space
\[\ker(\delta)^\perp = \{T \in \mathcal{S}: T = \mathcal{L}_vg\text{ for some geodesic vector field }v\}.\]
The notion of geodesic vector field on a Riemannian manifold \((M,g)\) was defined by \textit{K. Yano} and \textit{T. Nagano} in [C. R. Acad. Sci. Paris 252, 504--505, (1961; Zbl 0100.35903)] as a vector field \(v=(v^i)\) satisfying the condition \[ g^{ji}\Delta_j\Delta_iv^k +\mathrm{Ric}^k_iv^i = 0. \] In [\textit{K. Yano} and \textit{T. Nagano}, ``On geodesic vector fields in a compact orientable Riemannian space'', Comm. Math. Helv. 35, 55--64 (1961)] the authors proved the nonexistence of nonzero geodesic vector fields on a compact Riemannian manifold with negative-definite Ricci tensor. They showed that on an Einstein manifold \((M, g)\) with \(\mathrm{Ric} = cg\) the divergence of a geodesic vector field is a solution of the equation \(\Delta f = 2cf\) and a geodesic vector field is decomposed uniquely as the sum of a Killing vector field and a solution of the equation \(\Delta f = 2cf\).
In [the author and Nagano, loc. cit.] it was proved that the geodesic vector fields on a Riemannian manifold \((M, g)\) are the infinitesimal generators of the harmonic metrics on \((M, g)\). The authors showed that on a compact Kähler manifold \((M, J, g)\) a vector field \(v\) is a holomorphic vector field if and only if \(g'= \mathcal{L}_vg\) is a harmonic tensor on \((M, J, g)\). Moreover, they proved that if \(g'\) is a conservative harmonic metric of \((M, g)\) such that \(\mathrm{tr}_gg'=\mathrm{tr}_gg\), then the identity map \(i_M : (M, g) \rightarrow (M, g')\) is volume-decreasing, i.e., \(dv_g'\leq dv_g\) at each point of \(M\), and \(dv_{g'} = dv_g\) if and only if \(g' = g\) on \(M\).
For harmonic metrics on Kähler and hyper-Kähler manifolds the author quotes some results proved by \textit{Y. Watanabe} and \textit{R. Dohira} in [Math. J. Toyama Univ. 18, 137--146 (1995; Zbl 0848.53013)] and by \textit{A. D. Vîlcu} and \textit{G. E. Vîlcu} in [JP J. Geom. Topol. 7, 397--403 (2007; Zbl 1136.53318)].
In the previously cited paper by the author and Nagano [loc. cit.], it was shown that an isometric immersion \(\phi : (M, g_0) \rightarrow(E^m, \bar{g}_0)\) of a surface into a Euclidean space is harmonic if and only if \(\phi : (M, G_0) \rightarrow(E^m, \bar{g}_0)\) is harmonic, where \(G_0\) is the metric induced on \(M\) by the Gauss map \(\nu\) associated with \(\phi\). Subsequently, the Gauss map \(\nu\) is harmonic and the identity map \(\mathrm{id}_M:(M,G_0)\rightarrow (M,g_0)\) is homothetic if and only if \((M, g_0)\) has constant Gauss curvature or equivalently, there exists a hypersphere \(S^{m-1}\) of \(E^m\) containing \(\phi(M)\) such that \(\phi:(M, g_0) \rightarrow S^{m-1}\) is a harmonic map. This condition is equivalent to the harmonicity of the map \(\phi : (M, G_0) \rightarrow (E^{n+1}, \bar{g}_0)\) and it characterizes the situation when the identity map is conformal and the Gauss map is harmonic. If the identity map is an affine map, then either it is homothetic or the surfaces \((M, g_0)\) and \((M, G_0)\) are flat. Moreover, the identity map is an affine map and the Gauss map associated with \(\phi\) is harmonic if and only if \((M, g_0)\) has constant Gauss curvature and either \((M, g_0)\) is immersed in a hypersphere of \(E^m\) as a minimal surface via \(\phi\), or \((M, g_0)\) is immersed as an open subset of the product surface of two planar circles via \(\phi\).
In the book [the author and \textit{L. Verstraelen}, Laplace transformations of submanifolds. Leuven, Belgium: Katholieke Universiteit (1995; Zbl 0912.53036)], the authors considered an isometric immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\) of a Riemannian \(n\)-manifold into Euclidean \(m\)-space and studied the conditions under which the Laplace transformation \(\mathrm{id}_M:(M,g_0)\rightarrow(M, g_{L_\phi})\) determines the immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\), where \(L_\phi\) is the Laplace map and \(g_{L_\phi}\) is the metric induced via \(L_\phi\). The main results of this book are resumed in a section of the paper under review. Another section is devoted to the results on the Laplace-Gauss identity map (or LG-identity map) obtained in the same book. The property of the LG-identity map to be conformal or homothetic is related to the geometry of the manifold \(M\) and to the properties of the isometric immersion \(\phi:(M,g_0)\rightarrow (E^m, \bar{g}_0)\).
Regarding the tangent bundle of a pseudo-Riemannian manifold \((M,g)\), the author recalls that the identity map \(\mathrm{id}_{TM} : (TM, G) \rightarrow(TM, g^c)\) is biharmonic (i.e., the complete lift \(g^c\) of the metric \(g\) is biharmonic with respect to the Sasaki metric \(G\)) and that it is totally geodesic if and only if the projection \(\pi:(TM,G)\rightarrow(M,g)\) has the same property (see [\textit{C. Oniciuc}, Bull. Belg. Math. Soc. Simon Stevin 7, 443--454 (2000; Zbl 0983.53042)]). The author also recalls that in the more general case when \(G\) is an arbitrary \(g\)-natural metric on \(TM\), the identity map \(\mathrm{id}_{TM} : (TM, g^S) \rightarrow(TM,G)\) is harmonic if and only if a certain relation between some functions appearing in the expression of the metric \(G\) is satisfied and either the horizontal and vertical distributions are orthogonal with respect to \(G\), or \((M,g)\) is an Einstein manifold and a relation between two functions involved in the expression of \(G\) holds. Another characterization for the harmonicity of \(\mathrm{id}_{TM}\) consists in two relations satisfied by the functions from the expression of \(G\) (see [\textit{M.T.K. Abbassi} and \textit{G. Calvaruso}, Rend. Semin. Mat. Univ. Politec. Torino 68, 37--56 (2010; Zbl 1202.58011)]). For the case when \(g\) and \(\hat{g}\) are \(G\)-invariant pseudo-Riemannian metrics on a nonreductive homogeneous 4-manifold \(M^4\), it was proved in [\textit{A. Zaeim} and \textit{P. Atashpeykar}, Czechoslovak Math. J. 68(143), 475--490 (2018; Zbl 1488.53192)] that a pseudo-Riemannian metric \(\hat{g}\) is a harmonic metric on \((M^4, g)\) and the identity map of \(TM^4\) endowed with Sasaki metric, horizontal and complete lifts of the metrics \(g\) and \(\hat{g}\) is harmonic. In [\textit{C. L. Bejan} and \textit{S. L. Druță-Romaniuc}, Mediterr. J. Math. 12, 481--496 (2015; Zbl 1322.53065)] the authors proved the equivalence between the harmonicity of a Walker metric \(\hat{g}\) on a 4-dimensional Walker manifold \((W^4,g,\mathcal{D})\) and the harmonicity of the Sasaki metric \(\hat{g}^S\) and of the horizontal lift \(\hat{g}^h\) with respect to \({g}^S\) and \({g}^h\), respectively. Similar results for the harmonicity of Gödel-type metrics on Gödel-type spacetimes were obtained in [\textit{A. Zaeim} et al., Int. J. Geom. Methods Mod. Phys. 17, Article 2050092, 16 p. (2020)].
A (relative) biharmonic metric with respect to another metric \(g\) was defined in [\textit{P. Baird} and \textit{D. Kamissoko}, Ann. Global Anal. Geom. 23, 65--75 (2003; Zbl 1027.31004)] as a metric \(\bar{g}\) for which the identity map \(\mathrm{id}_M:(M, g) \rightarrow (M, \bar{g})\) is biharmonic. In the paper under review the author mentions some examples of proper biharmonic metrics constructed in his joint work with \textit{Y.-L. Ou} [Biharmonic submanifolds and biharmonic maps in Riemannian geometry, Hackensack, NJ: World Scientific (2020; Zbl 1455.53002)]. He also recalls that on a compact Einstein \(m\)-manifold \((M, g)\), for which \(\mathrm{Ric} = c g\), with \(m \geq 6\) and \(c < 0\), or with \(m > 6\) and \(c \leq 0\), the metric \(g' = e^{2\rho}g\) is biharmonic if and only if it is a harmonic metric for \((M, g)\).
The author quotes some references for harmonic metrics and compact symmetric spaces, stability of other spaces, Laplace transformations of submanifolds, harmonic metrics on generalized Sasakian space forms and on light-like manifolds or semi-Riemannian manifolds.
At the end of the paper, the author defines the notion of \(p\)-harmonic metric on a Riemannian manifold \((M,g)\) in an analogous way as for harmonic and biharmonic metrics, by the \(p\)-harmonicity of the identity map \(\mathrm{id}_M : (M, g) \rightarrow (M, g')\) and he proposes some open problems concerning the relations between \(p\)-harmonic metrics and \(k\)-harmonic metrics on a Riemannian manifold \((M, g)\) for \(k < p\).
For the entire collection see [Zbl 1490.53003].
Reviewer: Simona Druta-Romaniuc (Iaşi)Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditionshttps://zbmath.org/1521.530492023-11-13T18:48:18.785376Z"Braun, Mathias"https://zbmath.org/authors/?q=ai:braun.mathiasSummary: For a Lorentzian space measured by \(\mathfrak{m}\) in the sense of [\textit{M. Kunzinger} and \textit{C. Sämann}, Ann. Global Anal. Geom. 54, No. 3, 399--447 (2018; Zbl 1501.53057); \textit{F. Cavalletti} and \textit{A. Mondino}, Gen. Relativ. Gravitation 54, No. 11, Paper No. 137, 39 p. (2022; Zbl 1518.83004)], we introduce and study synthetic notions of timelike lower Ricci curvature bounds by \(K\in\mathbb{R}\) and upper dimension bounds by \(N\in[1,\infty)\), namely the timelike curvature-dimension conditions \(\mathrm{TCD}_p(K,N)\) and \(\mathrm{TCD}_p^\ast(K,N)\) in weak and strong forms, where \(p\in(0,1)\), and the timelike measure-contraction properties \(\mathrm{TMCP}(K,N)\) and \(\mathrm{TMCP}^\ast(K,N)\). These are formulated by convexity properties of the Rényi entropy with respect to \(\mathfrak{m}\) along \(\ell_p\)-geodesics of probability measures. We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological \(\ell_p\)-optimal couplings and chronological \(\ell_p\)-geodesics. We also prove the equivalence of \(\mathrm{TCD}_p^\ast(K,N)\) and \(\mathrm{TMCP}^\ast(K,N)\) to their respective entropic counterparts in the sense of [Cavalletti and Mondino, loc. cit.]. Some of these results are obtained under timelike \(p\)-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.Palatini variation in generalized geometry and string effective actionshttps://zbmath.org/1521.530572023-11-13T18:48:18.785376Z"Jurčo, Branislav"https://zbmath.org/authors/?q=ai:jurco.branislav"Moučka, Filip"https://zbmath.org/authors/?q=ai:moucka.filip"Vysoký, Jan"https://zbmath.org/authors/?q=ai:vysoky.janThe authors develop the Palatini formalism within the framework of a generalized Riemannian geometry of Courant algebroids. In the approach known as the Palatini variation, the action corresponding to field equations is a spacetime integral of the scalar curvature. In this context, one had to consider a general torsion-free affine connection. Let \(M\) be a Riemannian manifold with a metric \(g\). First, the authors introduce the required tools. Then, they recall the notions of a Courant algebroid, generalized metrics and courant algebroid connections. They show how ordinary affine connections can naturally be generalizd to Courant algebroid connections. Then, they extend the Palatini method to the setting of the generalized Riemannian geometry of Courant algebroids. The proof of the main results is given in detail and examples are also discussed.
Reviewer: Angela Gammella-Mathieu (Metz)Comparison of Poisson structures on moduli spaceshttps://zbmath.org/1521.530602023-11-13T18:48:18.785376Z"Biswas, Indranil"https://zbmath.org/authors/?q=ai:biswas.indranil"Bottacin, Francesco"https://zbmath.org/authors/?q=ai:bottacin.francesco"Gómez, Tomás L."https://zbmath.org/authors/?q=ai:gomez.tomas-lLet \(X\) be an irreducible smooth complex curve of genus \(g\). Consider a fixed algebraic line bundle \(N\) on \(X\). Let \(E\) be an algebraic vector bundle of rank \(r\) and degree \(\delta\), and \(\theta \in H^{0}(X, \mathcal{End}(E)\otimes N)\). A Hitchin pair \((E, \theta)\) is called stable if
\[
\deg F\cdot r < \delta\cdot \mathrm{rk} F
\]
for all nonzero proper subbundles \(F\subset E\) for which \(\theta(F) \subset F\otimes E\).
Let \(\mathcal{M}\) denote the moduli space of stable Hitchin pairs. It is known that \(\mathcal{M}\) has a nontrivial algebraic Poisson structure.
Let \(S\) be the smooth complex quasi-projective surface defined by the total space of the line bundle \(N\). Associated to \((E, \theta)\) there exist a subscheme \(Y_{(E, \theta)}\) and a coherent sheaf \(\mathcal{L}_{(E, \theta)}\). The pair \((Y_{(E, \theta)}, \mathcal{L}_{(E, \theta)})\) is known as the spectral datum of \((E, \theta)\). The spectral datum of a Hitchin pair produces an isomorphism \(\Phi:\mathcal{M}\to \mathcal{P}\) into the moduli space of stable sheaves of pure dimension \(1\) on \(S\).
A certain section \(\sigma\) of \(N\otimes K_{X}\) gives rise to a Poisson structure on the surface \(S\). Here \(K_{X}\) denotes as usual the canonical bundle of \(X\). An algebraic Poisson structure on \(\mathcal{P}\) is constructed with the Poisson structure on \(S\).
The main result of this paper is that the isomorphism \(\Phi\) sends the Poisson structure on \(\mathcal{M}\) to the Poisson structure on \(\mathcal{P}\). This generalizes previous results, for the case of the symplectic form of the moduli space of Higgs bundles, found in [\textit{I. Biswas} and \textit{A. Mukherjee}, Commun. Math. Phys. 221, No. 2, 293--304 (2001; Zbl 1066.14038); \textit{J. Harnad} and \textit{J. Hurtubise}, J. Math. Phys. 49, No. 6, 062903, 21 p. (2008; Zbl 1152.81465); \textit{J. C. Hurtubise} and \textit{M. Kjiri}, Commun. Math. Phys. 210, No. 2, 521--540 (2000; Zbl 0984.37090)].
Reviewer: Pablo Suárez-Serrato (Ciudad de México)A Leray-Serre spectral sequence for Lagrangian Floer theoryhttps://zbmath.org/1521.530642023-11-13T18:48:18.785376Z"Schultz, Douglas"https://zbmath.org/authors/?q=ai:schultz.douglasA symplectic Mori fibration is a fiber bundle of compact symplectic manifolds \((F,\omega_F)\to (E,\omega_{H,K})\stackrel{\pi}{\to}(B,\omega_B)\) whose transition maps are symplectomorphisms of the fibers, and where \((F,\omega_F)\) is monotone and \((B,\omega_B)\) is rational (i.e., \([\omega_B]\in H^2(B,\mathbb{Q})\)). We put \(\omega_{H,K} = a_H + K\pi^{\ast}\omega_B\) for large \(K\) with \(a_H\) the minimal coupling form associated to a connection \(H\) with Hamiltonian holonomy around contractible loops. Moreover, a fibered Lagrangian in a symplectic Mori fibration is a relatively spin Lagrangian \(L\subset (E,\omega_{H,K})\) such that
\[
L_{F}\to L\stackrel{\pi}{\to} L_{B},
\]
where \(L_F\subset (F,\omega_F)\) is a Lagrangian with minimal Maslov index two and \(L_B\subset (B,\omega_B)\) is a rational Lagrangian. The author uses as the coefficient ring a Novikov ring in two variables:
\[
\Lambda^{2}:= \left\{\sum\nolimits_{i,j} c_{ij}q^{\rho_{i}}r^{\eta_{j}}\mid c_{ij}\in\mathbb{C},\ \rho_{i}\geq 0,\ (1-\epsilon)\rho_{i} + \eta_{j}\geq 0,\ \#\{c_{ij}\neq 0, \ \rho_{i} + \eta_{j}\leq N\} < \infty \right\}.
\]
Then, the Floer complex \(CF(L,\Lambda^2)\) and the Floer ``differential'' \(\delta:CF(L,\Lambda^2)\to CF(L,\Lambda^2)\) are defined. Let
\[
\Lambda^{2}_{>0}:= \left\{\sum\nolimits_{i,j} c_{ij}q^{\rho_{i}}r^{\eta_{j}}\mid c_{ij}\in\mathbb{C},\ \rho_{i}\geq 0,\ (1-\epsilon)\rho_{i} + \eta_{j}> 0,\ \#\{c_{ij}\neq 0,\ \rho_{i} + \eta_{j}\leq N\} < \infty \right\}
\]
be the ring of elements with nonzero valuation in either \(q\) or \(r\). Since the definition of a symplectic Mori fibration is quite general, we will suppose that there is an element \(mc\in CF(L,\Lambda^{2}_{>0})\) of odd degree that solves the weak Maurer-Cartan equation, where the grading is defined. Thus \(\delta^{2}_{mc} = 0\), and we can define the Floer cohomology \(HF(L,\Lambda^{2},mc):= H(CF(L,\Lambda^{2},\delta_{mc}))\).
The author derives a Leray-Serre type spectral sequence to compute \(HF(L,\Lambda^2)\) by filtering the Floer chain complex considered above by \(q\)-degree or base energy.
Let us present the main results (skipping some technical details).
Theorem. Let \(F\to E\to B\) be a symplectic Mori fibration and \(L_{F}\to L\to L_{B}\) a fibered Lagrangian. Then:
(i) There is a coherent choice of perturbation data such that \(\delta\) is well defined;
(ii) Assuming that we have a solution \(mc\) to the weak Maurer-Cartan equation such that \(\delta^{2}_{mc} = 0\), there is a spectral sequence \(E^{\ast}_{s}\) that convergens to \(HF^{\ast}(L,\Lambda^{2})\).
Next, the author states a result about the second page, \(E_{2}^{\ast}\). Write
\[
\Lambda_r := \left\{\sum\nolimits_{j} c_{j}r^{\eta_{j}}\mid c_{j}\in\mathbb{C},\ \eta_{j}\geq 0,\ \#\{c_{j}\neq 0,\ \eta_{j}\leq N\} < \infty \right\}
\]
as the universal Novikov ring, and \(\delta_1\) its {vertical Floer differential.} Moreover, let \(mc_{0}\in CF(L,\Lambda_r)\) denote the part of \(mc\) obtained by setting \(q = 0\).
Theorem. The following statements hold true:
(i) \(\delta^{2}_{1,mc^{0}} = 0\), and \(E_{2}^{\ast}\simeq HF^{\ast}(L_F,E\mid_{L_{B}},mc^{0})\);
(ii) Assume the transition maps for \(E\) in a neighborhood of \(L_B\) take values in Ham\((F,\omega_F)\). Then \(mc^{0}_{M}\) is a weak Maurer-Cartan solution for \((F,L_F)\). If \(HF^{\ast}(L_F,F,\Lambda(t), mc^{0}_{M})\simeq 0\) then \(HF(L,E,\Lambda(t),mc)\simeq 0\), where we think of \(mc\) and \(mc_{M}^{0}\) as having coefficients from \(\Lambda(t)\) by replacing \(q\) and \(r\) with \(t\).
The author also gives an application to certain non-torus fibers of the Gelfand-Cetlin system in flag manifolds, and it is shown that their Floer cohomology vanishes.
Reviewer: Andrzej Szczepański (Gdańsk)Ricci flow on manifolds with boundary with arbitrary initial metrichttps://zbmath.org/1521.530712023-11-13T18:48:18.785376Z"Chow, Tsz-Kiu Aaron"https://zbmath.org/authors/?q=ai:chow.tsz-kiu-aaronThis paper is concerned with investigating the short time existence and uniqueness of the Ricci flow on a compact manifold \((M,g)\) with boundary \(\partial M\). That is, finding a solution to the following PDE with arbitrary smooth initial data \(g_0\):
\[
\begin{cases} \frac{\partial}{\partial t} g(t) = -2\mathrm{Ric}(g(t))&\text{on }M \times (0,T],\\
A_{g(t)} = 0 &\text{on } \partial M \times (0,T],\\
g(0) = g_0, \end{cases}
\]
where \(A_{g(t)}\) denotes the second fundamental form of the boundary \(\partial M\) with respect to the metric \(g(t)\). Past work on this problem required, for instance, that the boundary of the initial data \(g_0\) be totally geodesic. However, the author is able to remove this assumption.
The strategy of the proof is to consider the double \(\widetilde{M}\) of the compact manifold \(M\) with boundary. Since \(\widetilde{M}\) is a closed manifold, one would hope that standard results and techniques for Ricci flow could be used. However, the initial metric on \(\widetilde{M}\) is only Hölder continuous. In order to prove short-time existence and uniqueness for a Ricci flow starting at such rough initial data, the author uses a by now standard and powerful strategy of considering the Ricci-De Turck flow and then transferring the existence and uniqueness of such a flow to the desired Ricci flow. That each of these two flows induces a solution to the other additionally requires knowledge of a particular solution to the harmonic map heat flow. Given the roughness of the initial data in the aforementioned situations, a lot of detailed analysis, which includes dealing with suitable weighted parabolic Hölder spaces, is conducted by the author. This all culminates in uses of the Banach fixed point theorem to prove the desired results.
As a consequence of the work mentioned above, the author is able to prove that various curvature conditions are preserved along the Ricci flow, provided the boundary \(\partial M\) of the initial data satisfies certain curvature conditions. The list of these preserved conditions is the following:
(1) If \((M,g_0)\) has a convex boundary, then having positive curvature operator, being PIC\(1\), and being PIC\(2\) are all preserved under the flow;
(2) If \((M,g_0)\) has a two-convex boundary, then having PIC is preserved under the flow;
(3) If \((M,g_0)\) has mean convex boundary, then positive scalar curvature is preserved along the flow.
Reviewer: Louis Yudowitz (Stockholm)Regularity and curvature estimate for List's flow in four dimensionhttps://zbmath.org/1521.530742023-11-13T18:48:18.785376Z"Wu, Guoqiang"https://zbmath.org/authors/?q=ai:wu.guoqiangThis paper is devoted to List's flow, which is a triple $(M,g(t),\varphi (t))_{t\in (0,T)}$ satisfying
\begin{align*}
\partial_{t}g(t) &=-2\text{Ric}(g(t))+2d\varphi (t)\otimes d\varphi (t), \\
\partial_{t}\varphi (t) &=\Delta_{g(t)}\varphi (t), \\
g(0) &=g_{0},\quad\varphi (0)=\varphi_{0},
\end{align*}
where $\varphi (t):M\rightarrow \mathbb{R}$ are smooth functions, $g_{0}$ is a fixed Riemannian metric, and $\varphi_{0}$ is a fixed smooth function on a compact $n$-dimensional Riemannian manifold $M$ without boundary. Before the work by \textit{B. List} [Commun. Anal. Geom. 16, No. 5, 1007--1048 (2008; Zbl 1166.53044)], the Ricci flow system for a Riemannian metric $\partial_{t}g=-2\mathrm{Ric}(g)$ has been used with great
success for the construction of canonical metrics on Riemannian manifolds of low dimension. B. List has developed a corresponding theory for canonical objects with a certain physical interpretation. He proved the existence of an entropy $E$ such that the stationary points of List's flow are solutions to the static Einstein vacuum equations, and studied the extended parabolic system
\begin{align*}
\partial_{t}g &=-2\text{Ric}(g)+2\alpha_{n}d\varphi \otimes d\varphi , \\
\partial_{t}\varphi &=\Delta_{g}\varphi ,
\end{align*}
which is equivalent to the gradient flow of $E$. For applications on noncompact asymptotically flat manifolds, he proved short time existence on complete manifolds in the case when $\varphi $ is a smooth function from $M$ to $\mathbb{R}$.
In this paper the author studies List's flow on a compact manifold such that the scalar curvature is bounded. He establishes a time derivative bound for solutions to the heat equation, and derives the existence of a cutoff function (with good properties) whose time derivative and Laplacian are bounded. This can be seen as a parabolic version of Cheeger-Colding's cutoff function in the setting of Ricci lower bound.
Based on the above results, the author proves a backward pseudolocality theorem for the List's flow in dimension four. In this, the author needs to prove the $L^{\infty }$ estimate for subsolutions to nonhomogeneous linear heat equations along List's flow using Moser iteration method.
As an application, the author obtains that the $L^{2}$-norm of the Riemannian curvature operator is bounded and also gets the limit behavior of the List's flow. More precisely, based on an $L^{2}$-bound on the Riemannian curvature operator and a backward pseudolocality result, the author shows that if $M$ is a compact $4$-dimensional Riemannian manifold, $(M,g(t),\varphi (t)) $ is a List's flow on $[0,T)$, and if the trace of the Ricci tensor $S$ satisfies $|S|\leq 1$ on $M\times \lbrack 0,T),$ and $|\varphi_{0}|\leq 1$, then $(M,g(t),\varphi (t))$ converges to an orbifold in the Cheeger-Gromov sense as $t\rightarrow T$.
Reviewer: Boubaker-Khaled Sadallah (Algier)The Kähler-Ricci flow with log canonical singularitieshttps://zbmath.org/1521.530752023-11-13T18:48:18.785376Z"Chau, Albert"https://zbmath.org/authors/?q=ai:chau.albert"Ge, Huabin"https://zbmath.org/authors/?q=ai:ge.huabin"Li, Ka-Fai"https://zbmath.org/authors/?q=ai:li.ka-fai"Shen, Liangming"https://zbmath.org/authors/?q=ai:shen.liangmingSingularities for the Kähler-Ricci flow may develop in finite time even when starting with a nonsingular variety. The authors establish the existence of the Kähler-Ricci flow in the case of \(\mathbb Q\)-factorial projective varieties with log canonical singularities. This generalizes some of the existence results of \textit{J. Song} and \textit{G. Tian} [Invent. Math. 207, No. 2, 519--595 (2017; Zbl 1440.53116)] in the case of projective varieties with klt singularities. They also prove that the normalized Kähler-Ricci flow converges to a Kähler-Einstein metric with negative Ricci curvature on semi-log canonical models in the sense of currents. Further, they show that the weak Kähler-Ricci flow can be uniquely extended through the divisorial contractions and flips on \(\mathbb Q\)-factorial projective varieties with log canonical singularities.
Reviewer: Ljudmila Kamenova (New York)The Wecken problem for coincidences of boundary preserving surface mapshttps://zbmath.org/1521.550022023-11-13T18:48:18.785376Z"Kelly, Michael R."https://zbmath.org/authors/?q=ai:kelly.michael-r|kelly.michael-r-junThe goal of the work is to explore the non-Wecken property in the context of coincidence of relative maps. An interesting case is when \(X\) is a surface with non-empty boundary and the maps are relative maps \(f,g:(X, \partial X)\to (X, \partial X)\). The main result which is used to study the problem is an extension to the relative case of a result of \textit{R. B. S. Brooks} [Pac. J. Math. 40, 45--52 (1972; Zbl 0235.55006)] for the absolute case. Namely:
\textbf{Theorem 2.1}. Let \(X\) be a compact connected surface with non-empty boundary \(\partial X\). Let \(f, g : (X, \partial X) \to (X, \partial X)\) be boundary preserving maps and suppose that \(g\) is a homeomorphism. Then there is a boundary preserving map \(f_0\) homotopic to \(f\) such that \(\#\mathrm{Coin}(f_0, g) = \mu([f], [g])\).
Here \( \mu([f], [g])\) stands for the minimal number of coincidences among all pairs \((f', g')\) as \(f'\), \(g'\) run in the homotopy class \([f]\), \([g]\), respectively.
As an application of Theorem 2.1 above, making use of results of Nielsen fixed point theory of surface maps for the absolute case, the author shows for the relative case the following:
\textbf{Proposition 3.2}. Let \(F\) be a surface with non-empty boundary \(\partial F\) and negative Euler characteristic. Then there is a sequence of maps \(f_n\) such that
\((\mu([f_n], [g]) - N_{\mathrm{rel}}(f_n, g)) \to \infty\) as \(n \to \infty\). Here \(N_{rel}(f, g)\) stands for the relative Nielsen number of the pair \((f,g)\). The two cases of \(F\) not covered by the proposition are pointed out as interesting questions.
Reviewer: Daciberg Lima Gonçalves (São Paulo)Survey on \(L^2\)-invariants and 3-manifoldshttps://zbmath.org/1521.570152023-11-13T18:48:18.785376Z"Lück, Wolfgang"https://zbmath.org/authors/?q=ai:luck.wolfgangSummary: In this paper, we give a survey about \(L^2\)-invariants focusing on 3-manifolds.Two-fold refinement of non simply laced Chern-Simons theorieshttps://zbmath.org/1521.570272023-11-13T18:48:18.785376Z"Avetisyan, M. Y."https://zbmath.org/authors/?q=ai:avetisyan.m-y"Mkrtchyan, R. L."https://zbmath.org/authors/?q=ai:mkrtchyan.ruben-lSummary: Inspired by the two-parameter Macdonald's deformation of the formulae for non simply laced simple Lie algebras, we propose a two-fold refinement of the partition function of the corresponding Chern-Simons theory on \(S^3\). It is based on a two-fold refinement of the Kac-Peterson formula for the volume of the fundamental domain of the coroot lattice of non simply laced Lie algebras. We further derive explicit integral representations of the two-fold refined Chern-Simons partition functions. We also present the corresponding generalized universal-like expressions for them. With these formulae in hand, one can try to investigate a possible duality of the corresponding Chern-Simons theories with hypothetical two-fold refined topological string theories.Some useful operators on differential forms on Galilean and Carrollian spacetimeshttps://zbmath.org/1521.580012023-11-13T18:48:18.785376Z"Fecko, Marián"https://zbmath.org/authors/?q=ai:fecko.marianThis paper deals with some new operators and differential forms on Galilean and Carrollian spacetimes, where the theory of Hodge star operators is not well-established. The author shows that a Hodge star operator can be defined as an intertwining operator between representations on forms. As the author claims, this operator is a potential new tool for physicists working with Galilean and Carrollian spacetimes.
Reviewer: Alex B. Gaina (Chişinău)On the first eigenvalue of the Neumann problemhttps://zbmath.org/1521.580022023-11-13T18:48:18.785376Z"Meira, Adson"https://zbmath.org/authors/?q=ai:meira.adsonSummary: Let \(M^n\) be a smooth compact and connected Riemannian manifold with boundary \(\partial M \neq \emptyset\). Suppose \(M\) immersed into \(\mathbb{R}^{n+p}\), let \(\lambda_1 > 0\) be the first eigenvalue of the Neumann eigenvalue problem on \(M\), and \(H\) be the mean curvature of this immersion. Under a given condition, we prove that \(\lambda_1 vol(M) \leq n \int_M H^2 d V\), and the equality holds only if \(M\) is a minimal submanifold of some Euclidean hypersphere. This inequality was firstly proved by \textit{R. C. Reilly} [Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)]
for the closed eigenvalue problem.Non-degeneracy of critical points of the squared norm of the second fundamental form on manifolds with minimal boundaryhttps://zbmath.org/1521.580032023-11-13T18:48:18.785376Z"Cruz-Blázquez, Sergio"https://zbmath.org/authors/?q=ai:cruz-blazquez.sergio"Pistoia, Angela"https://zbmath.org/authors/?q=ai:pistoia.angelaAuthors' abstract: Let \((M, \overline{g})\) be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on \(\partial M\). We show that for a generic Riemannian metric \(\overline{g}\), the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate. We show that the generality of this property holds when we restrict ourselves to the conformal class of the initial metric on \(M\).
Reviewer: Marcelo Furtado (Brasília)An improved Morse index bound of min-max minimal hypersurfaceshttps://zbmath.org/1521.580042023-11-13T18:48:18.785376Z"Li, Yangyang"https://zbmath.org/authors/?q=ai:li.yangyangThe celebrated result [\textit{X. Zhou}, Ann. Math. (2), No. 3, 767--820 (2020; Zbl 1461.53051)] establishes an upper bound for minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold \(M^{n+1}\) with \(n+1\in[3,7]\). The present paper extends the result to any \(n+1\ge 3\) (ref.\ Theorem~1.3). The main techniques are the construction of hierarchical deformations, and a restrictive min-max theory not relying on bumpy metrics.
Reviewer: Giorgio Saracco (Firenze)Triharmonic CMC hypersurfaces in \({\mathbb{R}}^5(c)\)https://zbmath.org/1521.580052023-11-13T18:48:18.785376Z"Chen, Hang"https://zbmath.org/authors/?q=ai:chen.hang.1|chen.hang"Guan, Zhida"https://zbmath.org/authors/?q=ai:guan.zhidaSummary: A triharmonic map is a critical point of the tri-energy functional defined on the space of smooth maps between two Riemannian manifolds. In this paper, we prove that any CMC proper triharmonic hypersurface in the 5-dimensional space form \({\mathbb{R}}^5(c)\) must have constant scalar curvature. Furthermore, we show that any CMC triharmonic hypersurface in \({\mathbb{R}}^5\) or \({\mathbb{H}}^5\) must be minimal, which supports the generalized Chen's conjecture; we also give some characterizations of CMC proper triharmonic hypersurfaces in \({\mathbb{S}}^5\). Similar results are obtained in the higher dimension case under an additional assumption on the numbers of the distinct principal curvatures.Partial regularity for harmonic maps into spheres at a singular or degenerate free boundaryhttps://zbmath.org/1521.580062023-11-13T18:48:18.785376Z"Moser, Roger"https://zbmath.org/authors/?q=ai:moser.roger"Roberts, James"https://zbmath.org/authors/?q=ai:roberts.james-w|roberts.james-a|roberts.james-lMotivated by the study of fractional harmonic mappings, the authors provide regularity results for a class of free boundary harmonic mappings on certain domains exhibiting degenerate features.
Reviewer: Dumitru Motreanu (Perpignan)Diagrams and harmonic maps, revisitedhttps://zbmath.org/1521.580072023-11-13T18:48:18.785376Z"Pacheco, Rui"https://zbmath.org/authors/?q=ai:pacheco.rui"Wood, John C."https://zbmath.org/authors/?q=ai:wood.john-cIn the paper under review, the authors study and apply the finiteness criterion they have developed in collaboration with A. Aleman to extend many known results initially established for harmonic maps from the \(2\)-sphere into a Grassmannian
to harmonic maps from an arbitrary Riemann surface into a finite uniton number.
These results include: the description of harmonic maps of finite uniton number from any Riemann surface into a complex Grassmannian \(G_{2}(\mathbb{C}^{n})\). Noting that this extension relies on a new theory of nilpotent cycles. Also, a constancy result is developed showing that a harmonic map from a torus to a complex Grassmannian which is simultaneously of
finite uniton number and finite type is constant. Besides, authors generalize how harmonic maps of finite uniton number
can be constructed explicitly from holomorphic data from any Riemann surface instead of the \(2\)-sphere, using new methods.
Reviewer: Hiba Bibi (Tours)On the existence of solutions of a critical elliptic equation involving Hardy potential on compact Riemannian manifoldshttps://zbmath.org/1521.580082023-11-13T18:48:18.785376Z"Terki, Fatima Zohra"https://zbmath.org/authors/?q=ai:terki.fatima-zohra"Maliki, Youssef"https://zbmath.org/authors/?q=ai:maliki.youssefIn this paper, the authors study the existence of weak solutions to a certain non-linear elliptic equation on punctured compact Riemannian manifolds. This equation is a natural generalization of e.g. the equation for the scalar curvature in the Yamabe problem.
More precisely, let \((M,g)\) be an \(n\geq 3\) dimensional closed oriented Riemannian manifold with injectivity radius \(\delta_g>0\). Take a fixed point \(p\in M\) and define the truncated distance function \(\rho_p\) about this point to be the usual distance function in the geodesic ball \(B_{\delta_g}(p)\) and to be the constant \(\delta_g\) in the complementum \(M\setminus B_{\delta_g}(p)\). Let \(f,h\) be further smooth functions on \(M\) and consider the following nonlinear 2nd order scalar PDE on the punctured space \(M\setminus\{p\}\):
\[
\Delta_gu-\frac{h}{\rho_p^2}u=f\vert u\vert^{2^*-2}u
\]
where \(2^*=\frac{2n}{n-2}\) is the critical Sobolev exponent. Note that this equation gives back the Yamabe equation if the Hardy potential \(\frac{h}{\rho_p^2}\) is specified to \(\frac{n-2}{4(n-1)}\mathrm{Scal}_g\).
The authors study the existence of weak solutions to this equation over \(M\), more precisely solutions \(u\in L^2_1(M,g)\) or in different notation \(u\in H^2_1(M,g)\). For the precise technical statement cf. Theorems 1 and 2 in the paper.
Reviewer: Gábor Etesi (Budapest)An explicit Carleman formula for the Dolbeault cohomologyhttps://zbmath.org/1521.580092023-11-13T18:48:18.785376Z"Tarkhanov, Nikolai"https://zbmath.org/authors/?q=ai:tarkhanov.nikolai-nSummary: We study formulas which recover a Dolbeault cohomology class in a domain of \(\mathbb{C}^n\) through its values on an open part of the boundary. These are called Carleman formulas after the mathematician who first used such a formula for a simple problem of analytic continuation. For functions of several complex variables our approach gives the simplest formula of analytic continuation from a part of the boundary. The extension problem for the Dolbeault cohomology proves surprisingly to be stable at positive steps if the data are given on a concave piece of the boundary. In this case we construct an explicit extension formula.On the boundary complex of the \(k\)-Cauchy-Fueter complexhttps://zbmath.org/1521.580102023-11-13T18:48:18.785376Z"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.18Summary: The \(k\)-Cauchy-Fueter complex, \(k=0,1,\ldots\), in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy-Riemann complex in complex analysis. They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs-Bochner extension for \(k\)-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge-Ampère operator and open the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange's vanishing theorem and the Hartogs-Bochner extension for \(k\)-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.On invariants and equivalence of differential operators under Lie pseudogroups actionshttps://zbmath.org/1521.580112023-11-13T18:48:18.785376Z"Lychagin, Valentin"https://zbmath.org/authors/?q=ai:lychagin.valentin-v"Yumaguzhin, Valeriy"https://zbmath.org/authors/?q=ai:yumaguzhin.valeriy-aThe authors study invariants of linear differential operators with respect to algebraic Lie pseudogroups. They use these invariants and the principle of \(n\)-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, they use the pseudogroup of local symplectomorphisms.
Reviewer: Vladimir Balan (Bucureşti)Semiclassical asymptotic expansions for functions of the Bochner-Schrödinger operatorhttps://zbmath.org/1521.580122023-11-13T18:48:18.785376Z"Kordyukov, Y. A."https://zbmath.org/authors/?q=ai:kordyukov.yuri-aLet \((L,h^L)\) (resp. \((E,h^E)\)) be a Hermitian line (resp. vector) bundle on a \((X^d,g)\), a \(d\)-dimensional complete Riemannian manifold, and endowed with a Hermitian connection \(\nabla^L\) (resp. \(\nabla^E\)). The three objects \(X^d,L\), an \(E\) have the structure of bounded geometry, i.e., their associate curvatures, Levi-Civita connections and their covariant derivatives (of any order) are uniformly bounded, besides the injectivity radius of \((X^d,g)\) is positive. The Hermitian connection on \(L^p\otimes E\), the tensor product of \(L^p\) (the \(p\)th tensor power of \(L\)) with \(E\), is defined as \(\nabla ^{L^p\otimes E}\), the \(C^\infty(X^d,L^p\otimes E)\)-valued map on \(C^\infty(X^d,L^p\otimes E)\) which is induced by the Hermitian connections \(\nabla^L\) and \(\nabla^E\). The Bochner-Laplacian operator \(C^\infty(X^d,L^p\otimes E)\) is defined by \(\Delta^{L^p\otimes E}:=(\nabla ^{L^p\otimes E})^*\nabla ^{L^p\otimes E}\).
The author studies properties of the Bochner-Schrödinger operator \(H_p:= \frac{1}{p}\Delta^{L^p\otimes E}+V\) such that \(V\) is an \(End(E)\)-valued smooth potential on \(X^d\) where \(End(E)\) represents the space of endomorphisms on \(E\). To be precise, he provides a complete asymptotic expansion for Schwartz kernel of the operator \(\varphi(H_p)\) where \(\varphi\in\mathcal{S}(\mathbb R)\), the space of Schwartz functions on \(\mathbb R\).
Reviewer: Mohammed El Aïdi (Bogotá)On systoles and ortho spectrum rigidityhttps://zbmath.org/1521.580132023-11-13T18:48:18.785376Z"Masai, Hidetoshi"https://zbmath.org/authors/?q=ai:masai.hidetoshi"McShane, Greg"https://zbmath.org/authors/?q=ai:mcshane.gregA famous question asks if the spectrum of the Laplacian determines on a Riemann surface the geometry of the surface (``can one hear the shape of a drum?''). The answer to this question is no -- neither the length spectrum, nor the spectrum of the Laplacian determine the surface, but they do so up to finite ambiguity.
This article studies a question in the same circle of ideas. Namely, the authors consider a hyperbolic surface \(X\) with one geodesic boundary component, and ask to what extend the geometry is determined by the \textit{orthospectrum}, i.e. the lengths of all geodesics arcs meeting the boundary orthogonally.
It was known before that the length of the boundary, the area of the surface, and the entropy of the geodesic flow can be recovered [\textit{A. Basmajian}, Am. J. Math. 115, No. 5, 1139--1159 (1993; Zbl 0794.30032); \textit{M. Bridgeman}, Geom. Topol. 15, No. 2, 707--733 (2011; Zbl 1226.32007); \textit{S. P. Kerckhoff}, Ann. Math. (2) 117, 235--265 (1983; Zbl 0528.57008)]. The main theorems of this article show that the situation is the same as for the length spectrum: one can construct non-isometric surfaces with the same orthospectrum (Section 6), but there are only finitely many surfaces with the same orthospectrum (Theorem 5.1).
The examples in Section 6 use abelian covers to construct surfaces with the same orthospectrum, but different systoles. The proof of Theorem 5.1 follows a strategy by Wolpert, which proceeds by bounding the systole length from below using the orthospectrum. This is the part in which working with the orthospectrum is somewhat more subtle. The authors also include different arguments which only work for certain surfaces, which may nevertheless be of independent interest.
Reviewer: Sebastian Hensel (München)Bitangent planes of surfaces and applications to thermodynamicshttps://zbmath.org/1521.580142023-11-13T18:48:18.785376Z"Giblin, Peter"https://zbmath.org/authors/?q=ai:giblin.peter-j"Reeve, Graham"https://zbmath.org/authors/?q=ai:reeve.graham-markSummary: The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function \(A\) yield, for fixed temperature \(T\), a curve in the plane \(\mathbb{R}^2\) (one fluid) or a surface in 3-space \(\mathbb{R}^3\) (binary fluid). A line tangent to this curve in two places (bitangent line), or a set of planes tangent to this surface in two places (bitangent planes) have a thermodynamic significance which is well documented in the classical literature. Points of contact of bitangent planes trace `binodal curves' on the surface in \(\mathbb{R}^3\). The study of these bitangents is also classical, starting with D. J. Korteweg and J. D. van der Waals at the end of the \(19^{\mathrm{th}}\) century, but continuing into modern times. In this paper we give a summary of the thermodynamic background and of other mathematical investigations and then present a new mathematical approach which classifies a wide range of situations in \(\mathbb{R}^3\) where bitangents occur. In particular, we are able to justify many of the details in diagrams of binodal curves observed by Korteweg and others, using techniques from singularity theory.Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacianhttps://zbmath.org/1521.600462023-11-13T18:48:18.785376Z"Bañuelos, Rodrigo"https://zbmath.org/authors/?q=ai:banuelos.rodrigo"Mariano, Phanuel"https://zbmath.org/authors/?q=ai:mariano.phanuel-a"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.5Summary: For domains in \(\mathbb{R}^d\), \(d\geq 2\), we prove universal upper and lower bounds on the product of the bottom of the spectrum for the Laplacian to the power \(p>0\) and the supremum over all starting points of the \(p\)-moments of the exit time of Brownian motion. It is shown that the lower bound is sharp for integer values of \(p\) and that for \(p \geq 1\), the upper bound is asymptotically sharp as \(d\to \infty \). For all \(p>0\), we prove the existence of an extremal domain among the class of domains that are convex and symmetric with respect to all coordinate axes. For this class of domains we conjecture that the cube is extremal.Generalized stochastic areas, winding numbers, and hyperbolic Stiefel fibrationshttps://zbmath.org/1521.600472023-11-13T18:48:18.785376Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Demni, Nizar"https://zbmath.org/authors/?q=ai:demni.nizar"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing.5This paper fit into a larger research project concerning the study of integrable functionals of Brownian motions on symmetric spaces.
More recently, Brownian winding functionals were studied in several papers in the complex projective space and the complex hyperbolic space. Here, the Brownian motion is studied on the non-compact Grassmann manifold \(\frac{U(n-k,k)}{U(n-k)U(k)}.\) Realizing the Brownian motion on \(HG_{n,k}\) as a matrix diffusion process has the advantage to make available all the tools from stochastic calculus and random matrix theory.
Reviewer: Rózsa Horváth-Bokor (Budakalász)Riemannian Hamiltonian methods for min-max optimization on manifoldshttps://zbmath.org/1521.650512023-11-13T18:48:18.785376Z"Han, Andi"https://zbmath.org/authors/?q=ai:han.andi"Mishra, Bamdev"https://zbmath.org/authors/?q=ai:mishra.bamdev"Jawanpuria, Pratik"https://zbmath.org/authors/?q=ai:jawanpuria.pratik"Kumar, Pawan"https://zbmath.org/authors/?q=ai:kumar.pawan"Gao, Junbin"https://zbmath.org/authors/?q=ai:gao.junbinSummary: In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak-Łojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHMs) and present their convergence analyses. We extend RHMs to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHMs in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks.Stationary hypergeometric solitons and their stability in a Bose-Einstein condensate with \(\mathcal{PT}\)-symmetric potentialhttps://zbmath.org/1521.810622023-11-13T18:48:18.785376Z"Bhatia, Sanjana"https://zbmath.org/authors/?q=ai:bhatia.sanjana"Goyal, Amit"https://zbmath.org/authors/?q=ai:goyal.amit"Jana, Soumendu"https://zbmath.org/authors/?q=ai:jana.soumendu"Kumar, C. N."https://zbmath.org/authors/?q=ai:kumar.c-nagarajaSummary: We report the existence of stationary nonlinear matter-waves in a trapped Bose-Einstein condensate subject to a \(\mathcal{PT}\)-symmetric Pöschl-Teller potential with a gain/loss profile. Exact nonlinear modes are obtained and their stability criteria are determined. The analysis shows that beyond a critical depth of confining potential well, the condensate wavefunction is stable against small fluctuations in the field. Analytical results obtained are in good agreement with the numerical simulation of the localized modes in the \(\mathcal{PT}\) symmetry regime. Employing the isospectral hamiltonian technique of supersymmetric quantum mechanics, we demonstrate a mechanism to control the shape of the Pöschl-Teller well and hence the intensity of the localized modes. Most importantly, our results reveal that even with a small fluctuation present in the trapping potential bearing dissipation, the system is robust enough to support stable propagation of nonlinear modes.Caffarelli-Kohn-Nirenberg inequalities for curl-free vector fields and second order derivativeshttps://zbmath.org/1521.811202023-11-13T18:48:18.785376Z"Cazacu, Cristian"https://zbmath.org/authors/?q=ai:cazacu.cristian-m"Flynn, Joshua"https://zbmath.org/authors/?q=ai:flynn.joshua"Lam, Nguyen"https://zbmath.org/authors/?q=ai:lam.nguyenSummary: The present work has as a first goal to extend the previous results in [\textit{C. Cazacu} et al., J. Funct. Anal. 283, No. 10, Article ID 109659, 37 p. (2022; Zbl 1513.81089)] to weighted uncertainty principles with nontrivial radially symmetric weights applied to curl-free vector fields. Part of these new inequalities generalize the family of Caffarelli-Kohn-Nirenberg (CKN) inequalities studied by Catrina and Costa in [\textit{F. Catrina} and \textit{D. G. Costa}, J. Differ. Equations 246, No. 1, 164--182 (2009; Zbl 1220.35006)] from scalar fields to curl-free vector fields. We will apply a new representation of curl-free vector fields developed by \textit{N. Hamamoto} and \textit{F. Takahashi} [J. Funct. Anal. 280, No. 1, Article ID 108790, 24 p. (2021; Zbl 1452.26016)]. The newly obtained results are also sharp and minimizers are completely described. Secondly, we prove new sharp second order interpolation functional inequalities for scalar fields with radial weights generalizing the previous results in [Cazacu et al., loc. cit.]. We apply new factorization methods being inspired by our recent work [J. Differ. Equations 302, 533--549 (2021; Zbl 1499.81063)]. The main novelty in this case is that we are able to find a new independent family of minimizers based on the solutions of Kummer's differential equations. We point out that the two types of weighted inequalities under consideration (first order inequalities for curl-free vector fields vs. second order inequalities for scalar fields) represent independent families of inequalities unless the weights are trivial.Cyclic cocycles and one-loop corrections in the spectral actionhttps://zbmath.org/1521.811232023-11-13T18:48:18.785376Z"van Nuland, Teun D. H."https://zbmath.org/authors/?q=ai:van-nuland.teun-d-h"van Suijlekom, Walter D."https://zbmath.org/authors/?q=ai:van-suijlekom.walter-danielSummary: We present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd \((b,B)\)-cocycle, whereas, in the even orders, powers of the curvature are integrated against \((b,B)\)-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action \(\mathrm{Tr}(f(D+V))\) in powers of \(V=\pi_D(A)\), but unlike the Taylor expansion we expand in increasing order of the forms in \(A\). We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.
For the entire collection see [Zbl 1507.19001].Spectral properties of the symmetry generators of conformal quantum mechanics: a path-integral approachhttps://zbmath.org/1521.811322023-11-13T18:48:18.785376Z"Camblong, H. E."https://zbmath.org/authors/?q=ai:camblong.horacio-e"Chakraborty, A."https://zbmath.org/authors/?q=ai:chakraborty.abir|chakraborty.abhishek|chakraborty.avik|chakraborty.amit|chakraborty.arpan|chakraborty.ayan|chakraborty.archishman|chakraborty.anindita|chakraborty.ankur|chakraborty.aniruddha|chakraborty.abhijit|chakraborty.adrita|chakraborty.anujit|chakraborty.arunasis|chakraborty.aritra|chakraborty.apratim|chakraborty.aspriha|chakraborty.arindom|chakraborty.aruna|chakraborty.abhinav|chakraborty.ankit|chakraborty.arindam|chakraborty.ashit-b|chakraborty.anirvan|chakraborty.ashok-kr|chakraborty.amitava|chakraborty.amitabha|chakraborty.anwesha|chakraborty.amlan|chakraborty.arnab|chakraborty.atlanta|chakravorty.arnab-kumar|chakraborty.asa|chakraborty.anutosh|chakraborty.arup-k|chakraborty.arun|chakraborty.avishek|chakraborty.achin|chakraborty.aninda|chakraborty.antik|chakraborty.ajoy-kumar|chakraborty.adrijo|chakraborty.ananya|chakraborty.ashis-kumar|chakraborty.aishee"Lopez Duque, P."https://zbmath.org/authors/?q=ai:lopez-duque.p"Ordóñez, C. R."https://zbmath.org/authors/?q=ai:ordonez.carlos-rSummary: A path-integral approach is used to study the spectral properties of the generators of the \(\mathrm{SO}(2, 1)\) symmetry of conformal quantum mechanics (CQM). In particular, we consider the CQM version that corresponds to the weak-coupling regime of the inverse square potential. We develop a general framework to characterize a generic symmetry generator \(G\) (linear combinations of the Hamiltonian \(H\), special conformal operator \(K\), and dilation operator \(D\)), from which the path-integral propagators follow, leading to a complete spectral decomposition. This is done for the three classes of operators: Elliptic, parabolic, and hyperbolic. We also highlight novel results for the hyperbolic operators, with a continuous spectrum, and their quantum-mechanical interpretation. The spectral technique developed for the eigensystem of continuous-spectrum operators can be generalized to other operator problems.
{\copyright 2023 American Institute of Physics}Brane current algebras and generalised geometry from QP manifolds. Or, ``when they go high, we go low''https://zbmath.org/1521.812082023-11-13T18:48:18.785376Z"Arvanitakis, Alex S."https://zbmath.org/authors/?q=ai:arvanitakis.alex-sSummary: We construct a Poisson algebra of brane currents from a QP-manifold, and show their Poisson brackets take a universal geometric form. This generalises a result of Alekseev and Strobl on string currents and generalised geometry to include branes with worldvolume gauge fields, such as the D3 and M5. Our result yields a universal expression for the 't Hooft anomaly that afflicts isometries in the presence of fluxes. We determine the current algebra in terms of (exceptional) generalised geometry, and show that the tensor hierarchy gives rise to a brane current hierarchy. Exceptional complex structures produce pairs of anomaly-free current subalgebras on the M5-brane worldvolume.Duality cascades and parallelotopeshttps://zbmath.org/1521.812232023-11-13T18:48:18.785376Z"Furukawa, Tomohiro"https://zbmath.org/authors/?q=ai:furukawa.tomohiro"Moriyama, Sanefumi"https://zbmath.org/authors/?q=ai:moriyama.sanefumi"Sasaki, Hikaru"https://zbmath.org/authors/?q=ai:sasaki.hikaruSummary: Duality cascades are a series of duality transformations in field theories, which can be realized as the Hanany-Witten transitions in brane configurations on a circle. In the setup of the Aharony-Bergman-Jafferis-Maldacena theory and its generalizations, from the physical requirement that duality cascades always end and the final destination depends only on the initial brane configuration, we propose that the fundamental domain of supersymmetric brane configurations in duality cascades can tile the whole parameter space of relative ranks by translations, hence is a parallelotope. We provide our arguments for the proposal.Symmetric space \(\lambda\)-model exchange algebra from 4d holomorphic Chern-Simons theoryhttps://zbmath.org/1521.812352023-11-13T18:48:18.785376Z"Schmidtt, David M."https://zbmath.org/authors/?q=ai:schmidtt.david-mSummary: We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.One-loop partition function, gauge accessibility and spectra in \(\mathrm{AdS}_3\) gravityhttps://zbmath.org/1521.812762023-11-13T18:48:18.785376Z"Acosta, Joel"https://zbmath.org/authors/?q=ai:acosta.joel"Garbarz, Alan"https://zbmath.org/authors/?q=ai:garbarz.alan"Goya, Andrés"https://zbmath.org/authors/?q=ai:goya.andres"Leston, Mauricio"https://zbmath.org/authors/?q=ai:leston.mauricioSummary: We continue the study of the one-loop partition function of \(\mathrm{AdS}_3\) gravity with focus on the square-integrability condition on the fluctuating fields. In a previous work we found that the Brown-Henneaux boundary conditions follow directly from the \(L^2\) condition. Here we rederive the partition function as a ratio of Laplacian determinants by performing a suitable decomposition of the metric fluctuations. We pay special attention to the asymptotics of the fields appearing in the partition function. We also show that in the usual computation using ghost fields for the de Donder gauge, such gauge condition is accessible precisely for square-integrable ghost fields. Finally, we compute the spectrum of the relevant Laplacians in thermal \(\mathrm{AdS}_3\), in particular noticing that there are no isolated eigenvalues, only essential spectrum. This last result supports the analytic continuation approach of \textit{J. R. David} et al. [J. High Energy Phys. 2010, No. 4, Paper No. 125, 30 p. (2010; Zbl 1272.83081)]. The purely essential spectra found are consistent with the independent results of Lee and Delay of the essential spectrum of the TT rank-2 tensor Lichnerowickz Laplacian on asymptotically hyperbolic spaces.Relation between parity-even and parity-odd CFT correlation functions in three dimensionshttps://zbmath.org/1521.813122023-11-13T18:48:18.785376Z"Jain, Sachin"https://zbmath.org/authors/?q=ai:jain.sachin"John, Renjan Rajan"https://zbmath.org/authors/?q=ai:john.renjan-rajanSummary: In this paper we relate the parity-odd part of two and three point correlation functions in theories with exactly conserved or weakly broken higher spin symmetries to the parity-even part which can be computed from free theories. We also comment on higher point functions.
The well known connection of CFT correlation functions with de-Sitter amplitudes in one higher dimension implies a relation between parity-even and parity-odd amplitudes calculated using non-minimal interactions such as \(\mathcal{W}^3\) and \(\mathcal{W}^2\widetilde{\mathcal{W}}\). In the flat-space limit this implies a relation between parity-even and parity-odd parts of flat-space scattering amplitudes.A note on ensemble holography for rational torihttps://zbmath.org/1521.813262023-11-13T18:48:18.785376Z"Raeymaekers, Joris"https://zbmath.org/authors/?q=ai:raeymaekers.jorisSummary: We study simple examples of ensemble-averaged holography in free compact boson CFTs with rational values of the radius squared. These well-known rational CFTs have an extended chiral algebra generated by three currents. We consider the modular average of the vacuum character in these theories, which results in a weighted average over all modular invariants. In the simplest case, when the chiral algebra is primitive (in a sense we explain), the weights in this ensemble average are all equal. In the non-primitive case the ensemble weights are governed by a semigroup structure on the space of modular invariants.
These observations can be viewed as evidence for a holographic duality between the ensemble of CFTs and an exotic gravity theory based on a compact \(\mathrm{U}(1)\times\mathrm{U}(1)\) Chern-Simons action. In the bulk description, the extended chiral algebra arises from soliton sectors, and including these in the path integral on thermal \(\mathrm{AdS}_3\) leads to the vacuum character of the chiral algebra. We also comment on wormhole-like contributions to the multi-boundary path integral.Global anomalies and bordism invariants in one dimensionhttps://zbmath.org/1521.813592023-11-13T18:48:18.785376Z"Koizumi, Saki"https://zbmath.org/authors/?q=ai:koizumi.sakiSummary: We consider massless Majorana fermion systems with \(G = \mathbb{Z}_N\), \(SO(N)\), and \(O(N)\) symmetry in one-dimensional spacetime. In these theories, phase ambiguities of the partition functions are given as the exponential of the \(\eta\)-invariant of the Dirac operators in two dimensions, which is a bordism invariant. We construct sufficient numbers of bordism invariants to detect all bordism classes. Then, we classify global anomalies by calculating the \(\eta\)-invariant of these bordism classes.
{\copyright 2023 American Institute of Physics}Comments on the Atiyah-Patodi-Singer index theorem, domain wall, and Berry phasehttps://zbmath.org/1521.813622023-11-13T18:48:18.785376Z"Onogi, Tetsuya"https://zbmath.org/authors/?q=ai:onogi.tetsuya"Yoda, Takuya"https://zbmath.org/authors/?q=ai:yoda.takuyaSummary: It is known that the Atiyah-Patodi-Singer index can be reformulated as the eta invariant of the Dirac operators with a domain wall mass which plays a key role in the anomaly inflow of the topological insulator with boundary. In this paper, we give a conjecture that the reformulated version of the Atiyah-Patodi-Singer index can be given simply from the Berry phase associated with domain wall Dirac operators when adiabatic approximation is valid. We explicitly confirm this conjecture for a special case in two dimensions where an analytic calculation is possible. The Berry phase is divided into the bulk and the boundary contributions, each of which gives the bulk integration of the Chern character and the eta-invariant.The holographic contributions to the sphere free energyhttps://zbmath.org/1521.813742023-11-13T18:48:18.785376Z"Binder, Damon J."https://zbmath.org/authors/?q=ai:binder.damon-j"Freedman, Daniel Z."https://zbmath.org/authors/?q=ai:freedman.daniel-z"Pufu, Silviu S."https://zbmath.org/authors/?q=ai:pufu.silviu-s"Zan, Bernardo"https://zbmath.org/authors/?q=ai:zan.bernardoSummary: We study which bulk couplings contribute to the \(S^3\) free energy \(F(\mathfrak{m})\) of three-dimensional \(\mathcal{N} = 2\) superconformal field theories with holographic duals, potentially deformed by boundary real-mass parameters \(\mathfrak{m}\). In particular, we show that \(F(\mathfrak{m})\) is independent of a large class of bulk couplings that include non-chiral F-terms and all D-terms. On the other hand, in general, \(F(\mathfrak{m})\) does depend non-trivially on bulk chiral F-terms, such as prepotential interactions, and on bulk real-mass terms. These conclusions can be reached solely from properties of the AdS super-algebra, \(\mathfrak{osp}(2|4)\). We also consider massive vector multiplets in AdS, which in the dual field theory correspond to long single-trace superconformal multiplets of spin zero. We provide evidence that \(F(\mathfrak{m})\) is insensitive to the vector multiplet mass and to the interaction couplings between the massive vector multiplet and massless ones. In particular, this implies that \(F(\mathfrak{m})\) does not contain information about scaling dimensions or OPE coefficients of single-trace long scalar \(\mathcal{N} = 2\) superconformal multiplets.Modularity of supersymmetric partition functionshttps://zbmath.org/1521.813872023-11-13T18:48:18.785376Z"Gadde, Abhijit"https://zbmath.org/authors/?q=ai:gadde.abhijitSummary: We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional \(\mathcal{N} = 1\) supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.Subleading corrections to the \(S^3\) free energy of necklace quiver theories dual to massive IIAhttps://zbmath.org/1521.813912023-11-13T18:48:18.785376Z"Hong, Junho"https://zbmath.org/authors/?q=ai:hong.junho"Liu, James T."https://zbmath.org/authors/?q=ai:liu.james-tSummary: We investigate the \(S^3\) free energy of \(\mathcal{N} = 3\) Chern-Simons-matter quiver gauge theories with gauge group \(U(N)^r\) (\(r \geq 2\)) where the sum of Chern-Simons levels does not vanish, beyond the leading order in the large-\(N\) limit. We take two different approaches to explore the sub-leading structures of the free energy. First we evaluate the matrix integral for the partition function in the 't Hooft limit using a saddle point approximation. Second we use an ideal Fermi-gas model to compute the same partition function, but in the limit of fixed Chern-Simons levels. The resulting expressions for the free energy \(F = -\log Z\) are then compared in the overlapping parameter regime. The Fermi-gas approach also hints at a universal \(\frac{1}{6}\log N\) correction to the free energy. Since the quiver gauge theories we consider are dual to massive Type IIA theory, we expect the sub-leading correction of the planar free energy in the large 't Hooft parameter limit to match higher-derivative corrections to the tree-level holographic dual free energy, which have not yet been fully investigated.The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild backgroundhttps://zbmath.org/1521.830212023-11-13T18:48:18.785376Z"Hennig, Jörg"https://zbmath.org/authors/?q=ai:hennig.jorg-dieterSummary: We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the `cylinder', i.e. Friedrich's representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the \textit{Cauchy} problem for the conformally invariant wave equation \textit{J. Frauendiener} and \textit{J. Hennig} [Classical Quantum Gravity 35, No. 6, Article ID 065015, 19 p. (2018; Zbl 1386.83080)] showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at \(t = 0\). From a physical point of view, any data at \(t = 0\) are determined from the earlier history of the system and hence not exactly `free data'. Therefore, it is appropriate to ask what happens if we `go further back in time' and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding \textit{characteristic} initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders \(n = 0, \dots, 4\) for angular modes \(\ell = 0, \dots, 4\).Analysing (cosmological) singularity avoidance in loop quantum gravity using \(\mathrm{U}(1)^3\) coherent states and Kummer's functionshttps://zbmath.org/1521.830442023-11-13T18:48:18.785376Z"Giesel, Kristina"https://zbmath.org/authors/?q=ai:giesel.kristina"Winnekens, David"https://zbmath.org/authors/?q=ai:winnekens.davidSummary: Using a new procedure based on Kummer's Confluent Hypergeometric Functions, we investigate the question of singularity avoidance in loop quantum gravity (LQG) in the context of \(\mathrm{U}(1)^3\) complexifier coherent states and compare obtained results with already existing ones. Our analysis focuses on the dynamical operators, denoted by \(\hat{q}^{i_0}_{I_0}(r)\), whose products are the analogue of the inverse scale factor in LQG and also play a pivotal role for other dynamical operators such as matter Hamiltonians or the Hamiltonian constraint.
For graphs of cubic topology and linear powers in \(\hat{q}^{i_0}_{I_0}(r)\), we obtain the correct classical limit and demonstrate how higher order corrections can be computed with this method. This extends already existing techniques in the way how the involved fractional powers are handled. We also extend already existing formalisms to graphs with higher-valent vertices. For generic graphs and products of \(\hat{q}^{i_0}_{I_0}(r)\), using estimates becomes inevitable and we investigate upper bounds for these semiclassical expectation values. Compared to existing results, our method allows to keep fractional powers involved in \(\hat{q}^{i_0}_{I_0}(r)\) throughout the computations, which have been estimated by integer powers elsewhere. Similar to former results, we find a non-zero upper bound for the inverse scale factor at the initial singularity. Additionally, our findings provide some insights into properties and related implications of the results that arise when using estimates and can be used to look for improved estimates.A scattering theory approach to Cauchy horizon instability and applications to mass inflationhttps://zbmath.org/1521.831362023-11-13T18:48:18.785376Z"Luk, Jonathan"https://zbmath.org/authors/?q=ai:luk.jonathan"Oh, Sung-Jin"https://zbmath.org/authors/?q=ai:oh.sung-jin"Shlapentokh-Rothman, Yakov"https://zbmath.org/authors/?q=ai:shlapentokh-rothman.yakovSummary: Motivated by the strong cosmic censorship conjecture, we study the linear scalar wave equation in the interior of subextremal strictly charged Reissner-Nordström black holes by analyzing a suitably defined ``scattering map'' at 0 frequency. The method can already be demonstrated in the case of spherically symmetric scalar waves on Reissner-Nordström: we show that assuming suitable \((L^2\)-averaged) upper and lower bounds on the event horizon, one can prove \((L^2\)-averaged) polynomial lower bound for the solution
\begin{itemize}
\item[(1)] on any radial null hypersurface transversally intersecting the Cauchy horizon, and
\item[(2)] along the Cauchy horizon toward timelike infinity.
\end{itemize}
Taken together with known results regarding solutions to the wave equation in the exterior, (1) above in particular provides yet another proof of the linear instability of the Reissner-Nordström Cauchy horizon. As an application of (2) above, we prove a conditional mass inflation result for a nonlinear system, namely the Einstein-Maxwell-(real)-scalar field system in spherical symmetry. For this model, it is known that for a generic class of Cauchy data \(\mathcal{G}\), the maximal globally hyperbolic future developments are \(C^2\)-future-inextendible. We prove that if a (conjectural) improved decay result holds in the exterior region, then for the maximal globally hyperbolic developments arising from initial data in \(\mathcal{G}\), the Hawking mass blows up identically on the Cauchy horizon.Quantum state reduction, and Newtonian twistor theoryhttps://zbmath.org/1521.831592023-11-13T18:48:18.785376Z"Dunajski, Maciej"https://zbmath.org/authors/?q=ai:dunajski.maciej"Penrose, Roger"https://zbmath.org/authors/?q=ai:penrose.rogerSummary: We discuss the equivalence principle in quantum mechanics in the context of Newton-Cartan geometry, and non-relativistic twistor theory.Far-from-equilibrium attractors with a realistic non-conformal equation of statehttps://zbmath.org/1521.832022023-11-13T18:48:18.785376Z"Alqahtani, Mubarak"https://zbmath.org/authors/?q=ai:alqahtani.mubarakSummary: Using anisotropic hydrodynamics, we examine the existence of early-time attractors of non-conformal systems undergoing Bjorken expansion. In the case of a constant mass, we find that the evolution of the scaled longitudinal pressure is insensitive to variations of initial conditions converging onto an early-time universal curve and eventually merging with the late-time Navier-Stokes attractor (the hydrodynamic attractor). On the other hand, the bulk and the shear viscous corrections do not show an early-time attractor behavior. These results are consistent with previous studies considering a constant mass. When a realistic equation of state is included in the dynamics with a thermal mass, we demonstrate for the first time the absence of strict late-time universal attractors. However, a semi-universal feature of the evolution at very late times remains.Canonical analysis for Chern-Simons modification of general relativityhttps://zbmath.org/1521.832072023-11-13T18:48:18.785376Z"Escalante, Alberto"https://zbmath.org/authors/?q=ai:escalante.alberto"Pantoja-González, J. Aldair"https://zbmath.org/authors/?q=ai:pantoja-gonzalez.j-aldairSummary: By using the Gitman-Lyakhovich-Tyutin canonical analysis for higher-order theories a four-dimensional Chern-Simons modification of general relativity is analyzed. The counting of physical degrees of freedom, the symmetries, and the fundamental Dirac brackets are reported. Additionally, we report the complete structure of the constraints and its Dirac algebra is developed.Oscillatory path integrals for radio astronomyhttps://zbmath.org/1521.850032023-11-13T18:48:18.785376Z"Feldbrugge, Job"https://zbmath.org/authors/?q=ai:feldbrugge.job"Pen, Ue-Li"https://zbmath.org/authors/?q=ai:pen.ueli"Turok, Neil"https://zbmath.org/authors/?q=ai:turok.neil-gSummary: We introduce a new method for evaluating the oscillatory integrals which describe natural interference patterns. As an illustrative example of contemporary interest, we consider astrophysical plasma lensing of coherent sources like pulsars and fast radio bursts in radio astronomy. Plasma lenses are known to occur near the source, in the interstellar medium, as well as in the solar wind and the earth's ionosphere. Such lensing is strongest at long wavelengths, hence it is generally important to go beyond geometric optics and into the full wave optics regime.
Our computational method is a spinoff of new techniques two of us, and our collaborators, have developed for defining and performing Lorentzian path integralswith applications in quantum mechanics, condensed matter physics, and quantum gravity. Cauchy's theorem allows one to transform a computationally fragile and expensive, highly oscillatory integral into an exactly equivalent sum of absolutely and rapidly convergent integrals which can be evaluated in polynomial time. We require only that it is possible to analytically continue the lensing phase, expressed in the integrated coordinates, into the complex domain. We give a first-principles derivation of the Fresnel-Kirchhoff integral, starting from Feynman's path integral for a massless particle in a refractive medium. We then demonstrate the effectiveness of our method by computing the detailed diffraction patterns of Thom's caustic catastrophes, both in their ``normal forms'' and within a variety of more realistic, local lens models, for all wavelengths. Our numerical method, implemented in a freely downloadable code, provides a fast, accurate tool for modeling interference patterns in radioastronomy and other fields of physics.