Recent zbMATH articles in MSC 58https://zbmath.org/atom/cc/582022-11-17T18:59:28.764376ZUnknown authorWerkzeugRecent progress in mathematicshttps://zbmath.org/1496.140032022-11-17T18:59:28.764376ZPublisher's description: This book consists of five chapters presenting problems of current research in mathematics, with its history and development, current state, and possible future direction. Four of the chapters are expository in nature while one is based more directly on research. All deal with important areas of mathematics, however, such as algebraic geometry, topology, partial differential equations, Riemannian geometry, and harmonic analysis. This book is addressed to researchers who are interested in those subject areas.
Young-Hoon Kiem discusses classical enumerative geometry before string theory and improvements after string theory as well as some recent advances in quantum singularity theory, Donaldson-Thomas theory for Calabi-Yau 4-folds, and Vafa-Witten invariants.
Dongho Chae discusses the finite-time singularity problem for three-dimensional incompressible Euler equations. He presents Kato's classical local well-posedness results, Beale-Kato-Majda's blow-up criterion, and recent studies on the singularity problem for the 2D Boussinesq equations.
Simon Brendle discusses recent developments that have led to a complete classification of all the singularity models in a three-dimensional Riemannian manifold. He gives an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3.
Hyeonbae Kang reviews some of the developments in the Neumann-Poincare operator (NPO). His topics include visibility and invisibility via polarization tensors, the decay rate of eigenvalues and surface localization of plasmon, singular geometry and the essential spectrum, analysis of stress, and the structure of the elastic NPO.
Danny Calegari provides an explicit description of the shift locus as a complex of spaces over a contractible building. He describes the pieces in terms of dynamically extended laminations and of certain explicit ``discriminant-like'' affine algebraic varieties.
The articles of this volume will be reviewed individually.Positivity of vector bundles and Hodge theoryhttps://zbmath.org/1496.140102022-11-17T18:59:28.764376Z"Green, Mark"https://zbmath.org/authors/?q=ai:green.mark-lee"Griffiths, Phillip"https://zbmath.org/authors/?q=ai:griffiths.phillip-aPositivity and singularities are two of the central topics in complex algebraic/analytic geometry. From the viewpoint of differential geometry, these two properties are defined via the existence and behavior of certain types of metrics, which are usually characterized by their curvature properties.
In this mainly expository article, the authors give an account of some of the positivity notions of holomorphic vector bundles on complex manifolds: metric positivity (Griffiths or Nakano positivity), numerical positivity (defined in terms of positivity of Chern polynomials), and \textit{norm positivity} (which is an ``interpolation'' between the metric positivity and metric semi-positivity: see {Definition 2.13}) as well as their relations with the algebro-geometric positivity notions: nefness, bigness, and semi-ampleness.
When these bundles arise from Hodge theory, they have metrics induced from the polarization of the Hodge structures, and both the metric and numerical positivity are present and ultimately reflect the norm positivity property of the curvature of the Hodge bundles (see {Theorem 2.20}).
In Section 3 the authors first explain that the norm positive property can allow one to control the singularities of the Hodge bundles and as a result they show that the Hodge bundles have \textit{mild log singularities}. Then they discuss the \textit{monodromy weight filtrations} in the study of the asymptotic behaviors of several variable degenerations of Hodge structures. and they highlight the role of \textit{relative weight filtration property} in the analysis of singularities in the Chern form of the augmented Hodge bundle (see Section 3.2).
Reviewer: Jiaming Chen (Frankfurt am Main)The exponential map for Hopf algebrashttps://zbmath.org/1496.160332022-11-17T18:59:28.764376Z"Alhamzi, Ghaliah"https://zbmath.org/authors/?q=ai:alhamzi.ghaliah"Beggs, Edwin"https://zbmath.org/authors/?q=ai:beggs.edwin-jAuthors' abstract: We give an analogue of the classical exponential map on Lie groups for Hopf \(\star\)-algebras with differential calculus. The major difference with the classical case is the interpretation of the value of the exponential map, classically an element of the Lie group. We give interpretations as states on the Hopf algebra, elements of a Hilbert \(C^\star\)-bimodule of \(\frac{1}{2}\) densities and elements of the dual Hopf algebra. We give examples for complex valued functions on the groups \(S_3\) and \({\mathbb Z}\), Woronowicz's matrix quantum group \({\mathbb C}_q[SU_2]\) and the Sweedler-Taft algebra.
Reviewer: Salih Çelik (İstanbul)Local formulas for multiplicative formshttps://zbmath.org/1496.220012022-11-17T18:59:28.764376Z"Cabrera, A."https://zbmath.org/authors/?q=ai:cabrera.alejandro"Mărcuţ, I."https://zbmath.org/authors/?q=ai:marcut.ioan"Salazar, M. A."https://zbmath.org/authors/?q=ai:salazar.maria-ameliaIn a previous paper, the authors gave an explicit construction of a local Lie groupoid, for a given Lie algebroid, which is not necessarily integrable. In the paper under review, they extend this integration process to Lie algebroids endowed with extra infinitesimal data. This way, they give explicit formulas for the local version of various multiplicative forms on groupoids. Explicitly, they obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids.
Reviewer: Iakovos Androulidakis (Athína)Slice regular functions as covering maps and global \(\star\)-rootshttps://zbmath.org/1496.300282022-11-17T18:59:28.764376Z"Altavilla, Amedeo"https://zbmath.org/authors/?q=ai:altavilla.amedeo"Mongodi, Samuele"https://zbmath.org/authors/?q=ai:mongodi.samueleThe present work aims at studying slice regular functions of a quaternionic variable as covering maps and the existence and nature of global \(k\)-th \(\star\)-roots of slice functions. There are several steps done to reach the result. The authors introduce the formalism of stem functions and of slice preserving functions and their relations with the \(\star\)-product. They further show that, under suitable natural hypotheses, a slice regular function that is also a finite map is a covering map. Section 5 contains the main outcomes of this paper. On the basis of the results of the previous section, they are able to prove that, under suitable natural hypotheses, any slice regular function defined on a domain without real points admits \(k^2\) \(k\)-th \(\star\)-roots.
Reviewer: Swanhild Bernstein (Freiberg)Moduli of doubly connected domains under univalent harmonic mapshttps://zbmath.org/1496.310012022-11-17T18:59:28.764376Z"Bshouty, Daoud"https://zbmath.org/authors/?q=ai:bshouty.daoud-h"Lyzzaik, Abdallah"https://zbmath.org/authors/?q=ai:lyzzaik.abdallah"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Vasudevarao, Allu"https://zbmath.org/authors/?q=ai:vasudevarao.alluLet \(\mathcal{T}(t)=\mathbb{C}\setminus([-1,1]\cup[t,\infty))\), \(t>1\), be a Teichmüller doubly connected domain. The Teichmüller-Nitsche problem is formulated as follows:
For which values \(s,t\), \(1<s,t<\infty\), does a harmonic homeomorphism \(f:\mathcal{T}(s)\rightarrow\mathcal{T}(t)\) exist?
In the paper, the Teichmüller-Nitsche problem is solved for symmetric harmonic homeomorphisms between \(\mathcal{T}(s)\) and \(\mathcal{T}(t)\). This problem is solved by using the method of extremal length.
The following question suggested by \textit{T. Iwaniec} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 5, 1017--1030 (2011; Zbl 1267.30059)] is also considered:
Characterize pairs \((\Omega,\Omega^*)\) of doubly connected domains that admit a univalent harmonic mapping from \(\Omega\) onto \(\Omega^*\).
This question is tested regarding the moduli of the doubly connected domains related by harmonic homeomorphisms. The paper concludes with relevant questions.
Reviewer: Konstantin Malyutin (Kursk)On the dimension of Dolbeault harmonic \((1,1)\)-forms on almost Hermitian \(4\)-manifoldshttps://zbmath.org/1496.320412022-11-17T18:59:28.764376Z"Piovani, Riccardo"https://zbmath.org/authors/?q=ai:piovani.riccardo"Tomassini, Adriano"https://zbmath.org/authors/?q=ai:tomassini.adrianoSummary: We prove that the dimension \(h^{1,1}_{\overline{\partial}}\) of the space of Dolbeault harmonic \((1,1)\)-forms is not necessarily always equal to \(b^-\) on a compact almost complex \(4\)-manifold endowed with an almost Hermitian metric which is not locally conformally almost Kähler. Indeed, we provide examples of non integrable, non locally conformally almost Kähler, almost Hermitian structures on compact \(4\)-manifolds with \(h^{1,1}_{\overline{\partial}}=b^-+1\). This gives an answer to [Math. Z. 302, No. 1, 47--72 (2022; Zbl 1496.32035)] by \textit{T. Holt}.Super \(J\)-holomorphic curves: construction of the moduli spacehttps://zbmath.org/1496.320432022-11-17T18:59:28.764376Z"Keßler, Enno"https://zbmath.org/authors/?q=ai:kessler.enno"Sheshmani, Artan"https://zbmath.org/authors/?q=ai:sheshmani.artan"Yau, Shing-Tung"https://zbmath.org/authors/?q=ai:yau.shing-tungSummary: Let \(M\) be a super Riemann surface with holomorphic distribution \({\mathcal{D}}\) and \(N\) a symplectic manifold with compatible almost complex structure \(J\). We call a map \(\Phi :M\rightarrow N\) a super \(J\)-holomorphic curve if its differential maps the almost complex structure on \({\mathcal{D}}\) to \(J\). Such a super \(J\)-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super \(J\)-holomorphic curves as a smooth subsupermanifold of the space of maps \(M\rightarrow N\).The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampère flowshttps://zbmath.org/1496.320602022-11-17T18:59:28.764376Z"Guedj, Vincent"https://zbmath.org/authors/?q=ai:guedj.vincent"Lu, Chinh H."https://zbmath.org/authors/?q=ai:lu.chinh-h"Zeriahi, Ahmed"https://zbmath.org/authors/?q=ai:zeriahi.ahmedLet \(T>0\), \(\Omega\) be a bounded strictly pseudoconvex domain in \(\mathbb C^n\) and \(\Omega_T=(0,T)\times\Omega\). In this paper, the authors consider the following degenerate complex Monge-Ampère flow (CMAF), on \(\Omega_T\):
\[
dt\wedge(dd^cu)^n=e^{\partial_tu+F(t,z,u)}g(z)dt\wedge dV,
\]
where \(dV\) is the Euclidean volume on \(\mathbb C^n\), \(g\in L^p(\Omega)\) for some \(p>1\) and \(g>0\) a.e., \(F(t,z,r)\) is continuous on \([0,T)\times\Omega\times\mathbb R\) and bounded on \([0,T)\times\Omega\times J\) for each compact \(J\subset\mathbb R\), \(F\) is increasing in \(r\) and \((t,r)\to F(t,\cdot,r)\) is uniformly Lipschitz and semi-convex. This can be seen as a local version of the pluripotential Kähler-Ricci flow studied by the authors in the earlier work [Geom. Topol. 24, No. 3, 1225--1296 (2020; Zbl 1458.32035)].
To deal with the above equation (CMAF), the authors introduce and study the family \(\mathcal P(\Omega_T)\) of parabolic potentials. These are functions \(u:\Omega_T\to\mathbb R\cup\{-\infty\}\) whose slices \(u(t,\cdot)\) are plurisubharmonic and such that the family \(\{u(\cdot,z):\,z\in\Omega\}\) is locally uniformly Lipschitz in \((0,T)\). They prove in particular that the parabolic complex Monge-Ampère operator \(dt\wedge(dd^cu)^n\) is well defined as a measure for \(u\in\mathcal P(\Omega_T)\cap L^\infty_{\mathrm{loc}}(\Omega_T)\) and they introduce the notions of sub/super/solution to (CMAF) in this class. They also prove a comparison theorem for bounded parabolic potentials.
A function \(h\) defined on the parabolic boundary \(\big([0,T)\times\partial\Omega\big)\cup(\{0\}\times\Omega)\) of \(\Omega_T\) is a Cauchy-Dirichlet boundary data if \(h\) is continuous on \([0,T)\times\partial\Omega\), the family \(\{h(\cdot,z):\,z\in\partial\Omega\}\) is locally uniformly Lipschitz in \((0,T)\), \(h(0,\cdot)\in \operatorname{PSH}(\Omega)\cap L^\infty(\Omega)\) and \(\lim_{\Omega\ni z\to\zeta}h(0,z)=h(0,\zeta)\) for all \(\zeta\in\partial\Omega\).
The main result of the paper is that the Cauchy-Dirichlet problem for (CMAF) can be solved by the Perron method with parabolic potentials as admissible functions. Assuming that the Cauchy-Dirichlet boundary data \(h\) is such that for every \(S\in(0,T)\) there exists a constant \(C(S)\) with
\[
t|\partial_th(t,z)|\leq C(S) \text{ and } t^2\partial_t^2h(t,z)\leq C(S),\;\forall\,(t,z)\in (0,S]\times\partial\Omega,
\]
the authors prove that the pluripotential solution to this Cauchy-Dirichlet problem is given by the upper envelope of the family of all subsolutions with boundary data \(h\).
Reviewer: Dan Coman (Syracuse)Two solutions to Kirchhoff-type fourth-order implusive elastic beam equationshttps://zbmath.org/1496.340602022-11-17T18:59:28.764376Z"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.1"Yu, Wenguang"https://zbmath.org/authors/?q=ai:yu.wenguangSummary: In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.Fractional Laplacians : a short surveyhttps://zbmath.org/1496.350012022-11-17T18:59:28.764376Z"Daoud, Maha"https://zbmath.org/authors/?q=ai:daoud.maha"Laamri, El Haj"https://zbmath.org/authors/?q=ai:laamri.el-hajThe authors give an overview of the different operators which extend the Laplacian one to the fractional derivatives context. They concentrate on their very definitions and basic properties, stressing on some differences among them and the classical Laplacian, also by making use of explicit examples. Alongside, Sobolev spaces of fractional order are presented.
Reviewer: Nicola Abatangelo (Bologna)Patterns with prescribed numbers of critical points on topological torihttps://zbmath.org/1496.350602022-11-17T18:59:28.764376Z"Kamalia, Putri Zahra"https://zbmath.org/authors/?q=ai:kamalia.putri-zahra"Sakaguchi, Shigeru"https://zbmath.org/authors/?q=ai:sakaguchi.shigeruSummary: We study the existence of critical points of stable stationary solutions to reaction-diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with prescribed numbers of critical points whose locations are explicit.Convergence of the weighted Yamabe flowhttps://zbmath.org/1496.350942022-11-17T18:59:28.764376Z"Yan, Zetian"https://zbmath.org/authors/?q=ai:yan.zetianSummary: We introduce the weighted Yamabe flow
\[
\begin{cases}
\frac{\partial g}{\partial t} = (r_{\phi}^m - R_{\phi}^m) g \\
\frac{\partial \phi}{\partial t} = \frac{m}{2} (R_{\phi}^m - r_{\phi}^m)
\end{cases}
\]
on a smooth metric measure space \((M^n, g, e^{-\phi} \operatorname{dvol}_g, m)\), where \(R_{\phi}^m\) denotes the associated weighted scalar curvature, and \(r_{\phi}^m\) denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension \(n\) satisfies \(n \geqslant 3\).Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundaryhttps://zbmath.org/1496.351322022-11-17T18:59:28.764376Z"Hyder, Ali"https://zbmath.org/authors/?q=ai:hyder.ali"Segatti, Antonio"https://zbmath.org/authors/?q=ai:segatti.antonio"Sire, Yannick"https://zbmath.org/authors/?q=ai:sire.yannick"Wang, Changyou"https://zbmath.org/authors/?q=ai:wang.changyouSummary: We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.Uniform \(L^p\) resolvent estimates on the torushttps://zbmath.org/1496.351732022-11-17T18:59:28.764376Z"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathanThe author presents a new range of uniform \(L^p\) resolvent estimates, in the setting of the flat torus \({\mathbb T}^N:={\mathbb R}^N/{\mathbb Z}^N\) for \(N\ge 3\), improving previous results. Specifically, if \(\Delta_{{\mathbb T}^N}\) represents the Laplacian on the flat torus, then for all \(\varepsilon>0\) there exists a \(C_\varepsilon>0\) such that for all
\[
z\in\{z=(\lambda + i\mu)^2\in\mathbb{C}:\, \lambda, \mu\in\mathbb{R},\ \lambda\ge 1,\, |\mu|\ge \lambda^{-1/3+\varepsilon}\},
\]
the following holds
\[
\|u\|_{L^{2^*}({\mathbb T}^N)}\le C_\varepsilon\, \big\|(\Delta_{{\mathbb T}^N}+z)u\big\|_{L^{(2^*)'}({\mathbb T}^N)},
\]
where \(2^*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, and \((2^*)'=\frac{2N}{N+2}\) is its conjugate.
Their arguments rely on the \(l^2\)-decoupling theorem and multidimensional Weyl sum estimates.
Reviewer: Rosa Maria Pardo San Gil (Madrid)Nodal set of monochromatic waves satisfying the random wave modelhttps://zbmath.org/1496.351752022-11-17T18:59:28.764376Z"Romaniega, Álvaro"https://zbmath.org/authors/?q=ai:romaniega.alvaro"Sartori, Andrea"https://zbmath.org/authors/?q=ai:sartori.andreaSummary: We construct deterministic solutions to the Helmholtz equation in \(\mathbb{R}^m\) which behave accordingly to the Random Wave Model. We then find the number of their nodal domains, their nodal volume and the topologies and nesting trees of their nodal set in growing balls around the origin. The proof of the pseudo-random behaviour of the functions under consideration hinges on a de-randomisation technique pioneered by Bourgain and proceeds via computing their \(L^p\)-norms. The study of their nodal set relies on its stability properties and on the evaluation of their doubling index, in an average sense.The Birman-Krein formula for differential forms and electromagnetic scatteringhttps://zbmath.org/1496.351862022-11-17T18:59:28.764376Z"Strohmaier, Alexander"https://zbmath.org/authors/?q=ai:strohmaier.alexander"Waters, Alden"https://zbmath.org/authors/?q=ai:waters.aldenSummary: We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean near infinity. Allowing for compact boundaries of low regularity we prove a Birman-Krein formula on the space of co-closed differential forms. In the case of dimension three this reduces to a Birman-Krein formula in Maxwell scattering.A triviality result for semilinear parabolic equationshttps://zbmath.org/1496.352342022-11-17T18:59:28.764376Z"Castorina, Daniele"https://zbmath.org/authors/?q=ai:castorina.daniele"Catino, Giovanni"https://zbmath.org/authors/?q=ai:catino.giovanni"Mantegazza, Carlo"https://zbmath.org/authors/?q=ai:mantegazza.carloSummary: We show a triviality result for ``pointwise'' monotone in time, bounded ``eternal'' solutions of the semilinear heat equation \[u_t = \Delta u + |u|^p \] on complete Riemannian manifolds of dimension \(n \geq 5\) with nonnegative Ricci tensor, when \(p\) is smaller than the critical Sobolev exponent \(\frac{n+2}{n-2} \).Maximization of the second Laplacian eigenvalue on the spherehttps://zbmath.org/1496.352662022-11-17T18:59:28.764376Z"Kim, Hanna N."https://zbmath.org/authors/?q=ai:kim.hanna-nSummary: We prove a sharp isoperimetric inequality for the second nonzero eigenvalue of the Laplacian on \(S^m\). For \(S^2\), the second nonzero eigenvalue becomes maximal as the surface degenerates to two disjoint spheres, by a result of Nadirashvili for which Petrides later gave another proof. For higher dimensional spheres, the analogous upper bound was conjectured by Girouard, Nadirashvili and Polterovich. Our method to confirm the conjecture builds on Petrides' work and recent developments on the hyperbolic center of mass and provides also a simpler proof for \(S^2\).Sharp lower bound for the first eigenvalue of the weighted \(p\)-Laplacian. IIhttps://zbmath.org/1496.352672022-11-17T18:59:28.764376Z"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolong"Wang, Kui"https://zbmath.org/authors/?q=ai:wang.kuiSummary: Combined with our previous work [J. Geom. Anal. 31, No. 8, 8686--8708 (2021; Zbl 1475.35218)], we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted \(p\)-Laplacian with \(1<p<\infty\) on a compact Bakry-Émery manifold \((M^n,g,f)\), without boundary or with a convex boundary and Neumann boundary condition, satisfying \(\operatorname{Ric}+\nabla^2 f\geq \kappa g\) for some \(\kappa\in\mathbb{R}\).On the first eigenvalue of the Laplacian on compact surfaces of genus threehttps://zbmath.org/1496.352682022-11-17T18:59:28.764376Z"Ros, Antonio"https://zbmath.org/authors/?q=ai:ros.antonioSummary: For any compact Riemannian surface of genus three \((\Sigma, ds^2)\) Yang and Yau proved that the product of the first eigenvalue of the Laplacian \(\lambda_1(ds^2)\) and the area \(\mathit{Area}(ds^2)\) is bounded above by \(24\pi\). In this paper we improve the result and we show that \(\lambda_1(ds^2) \mathit{Area}(ds^2) \leq 16(4 - \sqrt{7})\pi \approx 21.668\pi\). About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value \(\approx 21.414\pi\).Generalized linking-type theorem with applications to strongly indefinite problems with sign-changing nonlinearitieshttps://zbmath.org/1496.353532022-11-17T18:59:28.764376Z"Bernini, Federico"https://zbmath.org/authors/?q=ai:bernini.federico"Bieganowski, Bartosz"https://zbmath.org/authors/?q=ai:bieganowski.bartoszSummary: We show a linking-type result which allows us to study strongly indefinite problems with sign-changing nonlinearities. We apply the abstract theory to the singular Schrödinger equation
\[
-\Delta u + V(x)u + \frac{a}{r^2} u = f(u) - \lambda g(u), \quad x = (y, z)\in\mathbb{R}^K \times\mathbb{R}^{N - K},\;r = |y|,
\]
where
\[
0 \notin \sigma\left(-\Delta + \frac{a}{r^2} + V(x)\right).
\]
As a consequence we obtain also the existence of solutions to the nonlinear curl-curl problem.Existence of ground state solutions for critical fractional Choquard equations involving periodic magnetic fieldhttps://zbmath.org/1496.354302022-11-17T18:59:28.764376Z"Jin, Zhen-Feng"https://zbmath.org/authors/?q=ai:jin.zhen-feng"Sun, Hong-Rui"https://zbmath.org/authors/?q=ai:sun.hongrui"Zhang, Jianjun"https://zbmath.org/authors/?q=ai:zhang.jianjun|zhang.jianjun.1Summary: In this paper, we consider the following critical fractional magnetic Choquard equation:
\[
\begin{aligned}
{\varepsilon }^{2s}{(-\Delta)}_{A/\varepsilon}^su+V(x)u \,= \; &{\varepsilon}^{\alpha-N} \left(\int_{\mathbb{R}^N} \frac{|u(y)|^{2_{s,\alpha}^\ast}}{|x-y|^\alpha}\mathrm{d}y\right) |u|^{2_{s,\alpha}^\ast-2}u \\
& +{\varepsilon }^{\alpha-N}\left(\int_{\mathbb{R}^N}\frac{F(y,| u(y)|^2)}{| x-y|^\alpha} \mathrm{d}y\right) f(x,|u|^2)u \quad\text{in }\mathbb{R}^N,
\end{aligned}
\] where \(\varepsilon > 0\), \(s\in (0,1)\), \(\alpha \in (0,N)\), \(N> \max \{2 \mu +4s,2s+\alpha /2\}\), \(2_{s,\alpha}^{\ast}=\frac{2N-\alpha}{N-2s}\) is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, \((-\Delta)_A^s\) stands for the fractional Laplacian with periodic magnetic field A of \(C^{0,\mu}\)-class with \(\mu \in (0,1]\) and \(V\) is a continuous potential and allows to be sign-changing. Under some mild assumptions imposed on \(V\) and \(f\), we establish the existence of at least one ground state solution.Existence results for nonlinear Schrödinger equations involving the fractional \((p, q)\)-Laplacian and critical nonlinearitieshttps://zbmath.org/1496.354352022-11-17T18:59:28.764376Z"Lv, Huilin"https://zbmath.org/authors/?q=ai:lv.huilin"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhou"Feng, Zhaosheng"https://zbmath.org/authors/?q=ai:feng.zhaoshengSummary: In this article, we consider the existence of ground state positive solutions for nonlinear Schrodinger equations of the fractional \((p, q)\)-Laplacian with Rabinowitz potentials defined in \(\mathbb{R}^n\),
\[(-\Delta)^{s_{1}}_p u+(-\Delta)^{s_{2}}_q u+V(\varepsilon x)(|u|^{p-2} u+|u|^{q-2} u)=\lambda f(u)+\sigma|u|^{q^{\ast}_{s_{2}}-2}u. \]
We prove existence by confining different ranges of the parameter \(\lambda\) under the subcritical or critical nonlinearities caused by \(\sigma =0\) or 1, respectively. In particular, a delicate calculation for the critical growth is provided so as to avoid the failure of a global Palais-Smale condition for the energy functional.Heat flow for harmonic maps from graphs into Riemannian manifoldshttps://zbmath.org/1496.370382022-11-17T18:59:28.764376Z"Baird, Paul"https://zbmath.org/authors/?q=ai:baird.paul"Fardoun, Ali"https://zbmath.org/authors/?q=ai:fardoun.ali"Regbaoui, Rachid"https://zbmath.org/authors/?q=ai:regbaoui.rachidSummary: We introduce the notion of harmonic map from a graph into a Riemannian manifold via a discrete version of the energy density. Existence and basic properties are established. Global existence and convergence of the associated heat flow are proved without any assumption on the curvature of the target manifold. We discuss a variant of the Steiner problem which replaces length by elastic energy.Lipschitz and Fourier type conditions with moduli of continuity in rank \(1\) symmetric spaceshttps://zbmath.org/1496.420042022-11-17T18:59:28.764376Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Restrepo, Joel E."https://zbmath.org/authors/?q=ai:restrepo.joel-esteban"Suragan, Durvudkhan"https://zbmath.org/authors/?q=ai:suragan.durvudkhanSummary: Sufficient and necessary results have been proven on Lipschitz type integral conditions and bounds of its Fourier transform for an \(L^2\) function, in the setting of Riemannian symmetric spaces of rank \(1\) whose growth depends on a \(k\)th-order modulus of continuity.A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\)https://zbmath.org/1496.430062022-11-17T18:59:28.764376Z"Bui, Huan Q."https://zbmath.org/authors/?q=ai:bui.huan-q"Randles, Evan"https://zbmath.org/authors/?q=ai:randles.evanSummary: In this article, we consider a class of functions on \(\mathbb{R}^d\), called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\). As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function \(P\), we construct a Radon measure \(\sigma_P\) on \(S=\{\eta \in\mathbb{R}^d:P(\eta)=1\}\) which is invariant under the symmetry group of \(P\). With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on \(\mathbb{Z}^d\) and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste
[\textit{E.~Randles} and \textit{L.~Saloff-Coste}, Rev. Mat. Iberoam. 33, No.~3, 1045--1121 (2017; Zbl 1377.42012)].Baum-Connes and the Fourier-Mukai transformhttps://zbmath.org/1496.460722022-11-17T18:59:28.764376Z"Emerson, Heath"https://zbmath.org/authors/?q=ai:emerson.heath"Hudson, Daniel"https://zbmath.org/authors/?q=ai:hudson.danielIf \(\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d\) is the torus, the Poincaré bundle \(\mathcal{P}_d\) is a complex line bundle over \(\mathbb{T}^d \times \widehat{\mathbb{Z}}^d\) (where \(\widehat{\mathbb{Z}}^d = \Hom(\mathbb{Z}^d,\mathbb{T})\)). The Fourier-Mukai correspondence is the topological correspondence
\[
\mathbb{T}^d \xleftarrow{\text{pr}_1} (\mathbb{T}^d\times \widehat{\mathbb{Z}}^d, \mathcal{P}_d) \xrightarrow{\text{pr}_2} \widehat{\mathbb{Z}}^d,
\]
which defines an element \([\mathcal{F}_d] \in KK_{-d}(\mathbb{T}^d, \widehat{\mathbb{Z}}^d)\). The Fourier-Mukai transform is the map \([\mathcal{F}_d]\otimes_{\widehat{\mathbb{Z}}^d} : K^{\ast}(\widehat{\mathbb{Z}}^d) \to\) \(K^{\ast -d}(\mathbb{T}^d)\), and the first main result of the paper is a geometric description of this map.
Given a torsion-free discrete group \(G\) with classifying space \(BG\), the Baum-Connes assembly map
\[
\mu : K_{\ast}(BG) \to K_{\ast}(C^{\ast}(G))
\]
is given by
\[
\mu(f) = \mathcal{P}_G\otimes_{C(G)\otimes C^{\ast}(G)}(f\otimes 1_{C^{\ast}(G)})
\]
where \(\mathcal{P}_G \in KK_0(\mathbb{C}, C(BG)\otimes C^{\ast}(G))\) is the class of the Mischenko element (see [\textit{P. Baum} et al., Contemp. Math. 167, 241--291 (1994; Zbl 0830.46061)]). If \(G = \mathbb{Z}^d\), then \(\mathcal{P}_G\) is the class of the Poincaré bundle and the second main result of the paper is a geometric description of the assembly map in this case.
Reviewer: Prahlad Vaidyanathan (Bhopal)The category of compact quantum metric spaceshttps://zbmath.org/1496.460742022-11-17T18:59:28.764376Z"Long, Botao"https://zbmath.org/authors/?q=ai:long.botao"Wu, Wei"https://zbmath.org/authors/?q=ai:wu.wei.3Attractors for semigroups and evolution equations. With an introduction by Gregory A. Seregin, Varga K. Kalantarov and Sergey V. Zelikhttps://zbmath.org/1496.470012022-11-17T18:59:28.764376Z"Ladyzhenskaya, Olga A."https://zbmath.org/authors/?q=ai:ladyzhenskaya.olga-aleksandrovnaSee the review of the 1991 edition in [Zbl 0755.47049].Minimal and maximal extensions of \(M\)-hypoelliptic proper uniform pseudo-differential operators in \(L^p\)-spaces on non-compact manifoldshttps://zbmath.org/1496.470762022-11-17T18:59:28.764376Z"Milatovic, Ognjen"https://zbmath.org/authors/?q=ai:milatovic.ognjenThe paper concerns pseudo-differential operators on manifolds of bounded geometry. In [\textit{Yu. A. Kordyukov}, Acta Appl. Math. 23, No. 3, 223--260 (1991; Zbl 0743.58030)] and [\textit{M. A. Shubin}, in: Méthodes semi-classiques. Vol. 1. École d'été (Nantes, juin 1991). Paris: Société Mathématique de France. 35--108 (1992; Zbl 0793.58039)], proper uniform elliptic operators were considered in this setting with action on the \(p\)-Lebesgue spaces. Willing to pass to more general pseudo-differential operators, the author here addresses to the classes of \textit{G. Garello} and \textit{A. Morando} [Integral Equations Oper. Theory 51, No. 4, 501--517 (2005; Zbl 1082.35175)]. For them, action on Lebesgue spaces is granted and the classical ellipticity condition is generalized by the notion of multi-quasi-ellipticity. The author introduces in this context a related definition of proper uniform operator on a manifold with bounded geometry. As an application, equality of minimal and maximal extension of the operator is proved. The paper is a very interesting contribution to the theory of pseudo-differential operators on non-compact manifolds. Similar results for the same classes in Euclidean spaces were proved by \textit{M. W. Wong} [Math. Nachr. 279, No. 3, 319--326 (2006; Zbl 1102.47038)] and \textit{V. Catană} [Appl. Anal. 87, No. 6, 657--666 (2008; Zbl 1158.47031)].
Reviewer: Luigi Rodino (Torino)On the first eigenvalue of the \(p\)-Laplacian on Riemannian manifoldshttps://zbmath.org/1496.530022022-11-17T18:59:28.764376Z"Seto, Shoo"https://zbmath.org/authors/?q=ai:seto.shooSummary: We survey results on the first (nontrivial) eigenvalue of the \(p\)-Laplace operator for both the Dirichlet and Neumann/closed condition on Riemannian manifolds. We also discuss an extension of the \(p\)-Laplace operator to act on differential forms. Some potential future directions of work are also given.
For the entire collection see [Zbl 1495.53004].The fundamental theorem of Legendrian submanifolds in the Heisenberg grouphttps://zbmath.org/1496.530222022-11-17T18:59:28.764376Z"Chiu, Hung-Lin"https://zbmath.org/authors/?q=ai:chiu.hung-lin"Lai, Sin-Hua"https://zbmath.org/authors/?q=ai:lai.sin-hua"Li, Jian-Wei"https://zbmath.org/authors/?q=ai:li.jianweiSummary: We study uniqueness and existence questions for Legendrian submanifolds in the Heisenberg group via Cartan's method of moving frames and the theory of Lie groups. Moreover, we generalize this result to any dimensional isotropic submanifolds in the Heisenberg group.The classifying Lie algebroid of a geometric structure. II: \(G\)-structures with connectionhttps://zbmath.org/1496.530312022-11-17T18:59:28.764376Z"Loja Fernandes, Rui"https://zbmath.org/authors/?q=ai:fernandes.rui-loja"Struchiner, Ivan"https://zbmath.org/authors/?q=ai:struchiner.ivanThis work is the second of two papers dedicated to a systematic study of symmetries, invariants and moduli spaces of geometric structures of finite type, and contains part of the reults obtained by the second author in his Ph.D. thesis [The classifiying Lie algebroid of a geometric structure. University of Campinas (PhD Thesis) (2009)].
The first paper [\textit{R. Loja Fernandes} and \textit{I. Struchiner}, Trans. Am. Math. Soc. 366, No. 5, 2419--2462 (2014; Zbl 1285.53018)] was dedicated to the case of \(\{e\}\)-structures and a special case of Cartan's realization problem. The central object introduced was the classifying Lie algebroid of a fully regular coframe. The classifying Lie algebroid contains all the relevant information for the equivalence problem for \(\{e\}\)-structures.
In the work under review the authors do similar analysis for general \(G\)-structures and the general case of Cartan's realization problem. The appropiate language to deal with such problem is the theory of \(G\)-structure groupoids and \(G\)-structure algebroids developed by the authors in [``The global solutions to Cartan's realization problem'', Preprint, \url{arXiv:1907.13614}].
Given a \(G\)-structure on a manifold \(M\), denoted by \(F_{G}(M)\), equipped with a connection \(\omega \in \Omega ^{1}(F_{G}(M),\mathfrak{g})\), where \( F_{G}(M)\) is the Lie algebra of \(G\), let \(\Omega ^{\bullet }(F_{G}(M), \mathfrak{g})\) be the space of invariants forms consisting of all differential forms which are preserved under local equivalences of \(\left( F_{G}(M),\omega \right) \). The \(G\)-structure with connection \(\left( F_{G}(M),\omega \right) \) is said to be fully regular when the space
\[
\left\{ d_{p}I:I\in \Omega ^{0}(F_{G}(M),\omega )\right\} \subset T_{p}^{\ast }F_{G}(M)\text{, }p\in F_{G}(M)
\]
has constant dimension. To fully regular \(G\)-structure with connection \(\left( F_{G}(M),\omega \right) \), there is naturally associated a vector bundle \(A\rightarrow X\) such that
\[
\Omega ^{\bullet }(F_{G}(M),\mathfrak{g})\simeq \Gamma (\wedge ^{\bullet }A^{\ast }).
\]
It follows that \(A\) has a Lie algebroid structure, called the classifying Lie algebroid of \(\left( F_{G}(M),\omega \right) \).
In this work the authors make precise the link between \(G\)-structure algebroids with connection and the classifying Lie algebroid of a single \(G\)-structure. In the last section, they show how \(G\)-realizations of a \(G\)-structure algebroid with connection are related to \(G\)-integration and how they can be used to codify the solutions of the associated realization method of Cartan.
Reviewer: Eugenia Rosado María (Madrid)Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifoldshttps://zbmath.org/1496.530482022-11-17T18:59:28.764376Z"Bray, Hubert L."https://zbmath.org/authors/?q=ai:bray.hubert-l"Kazaras, Demetre P."https://zbmath.org/authors/?q=ai:kazaras.demetre-p"Khuri, Marcus A."https://zbmath.org/authors/?q=ai:khuri.marcus-a"Stern, Daniel L."https://zbmath.org/authors/?q=ai:stern.daniel-lSummary: An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen-Yau minimal hypersurfaces.New integral estimates in substatic Riemannian manifolds and the Alexandrov theoremhttps://zbmath.org/1496.530562022-11-17T18:59:28.764376Z"Fogagnolo, Mattia"https://zbmath.org/authors/?q=ai:fogagnolo.mattia"Pinamonti, Andrea"https://zbmath.org/authors/?q=ai:pinamonti.andreaSummary: We derive new integral estimates on substatic manifolds with boundary of horizon type, naturally arising in General Relativity. In particular, we generalize to this setting an identity due to [\textit{R. Magnanini} and \textit{G. Poggesi}, J. Anal. Math. 139, No. 1, 179--205 (2019; Zbl 1472.53013)] leading to the Alexandrov Theorem in \(\mathbb{R}^n\) and improve on a Heintze-Karcher type inequality due to [\textit{J. Li} and \textit{C. Xia}, J. Differ. Geom. 113, No. 3, 493--518 (2019; Zbl 1433.53059)]. Our method relies on the introduction of a new vector field with nonnegative divergence, generalizing to this setting the P-function technique of \textit{H. F. Weinberger} [Arch. Ration. Mech. Anal. 43, 319--320 (1971; Zbl 0222.31008)].Closed geodesics on reversible Finsler 2-sphereshttps://zbmath.org/1496.530582022-11-17T18:59:28.764376Z"De Philippis, Guido"https://zbmath.org/authors/?q=ai:de-philippis.guido"Marini, Michele"https://zbmath.org/authors/?q=ai:marini.michele"Mazzucchelli, Marco"https://zbmath.org/authors/?q=ai:mazzucchelli.marco"Suhr, Stefan"https://zbmath.org/authors/?q=ai:suhr.stefanLet \((M,F)\) be a closed oriented surface endowed with a reversible Finsler metric \(F\), \(\mathbb S^1\) be the unit circle, and \(\mathrm{Emb}(\mathbb S^1,M)\) be the set of smooth \(M\)-valued embedded loops \(\gamma\) on \(\mathbb S^1\). Given \(J\in \mathrm{End}(TM)\), the positive normal to \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) is defined by \(N_{\gamma_t}(u)=\displaystyle\frac{J_{\dot{\gamma_t}}(u)}{\|\dot{\gamma_t}(u)\|}\cdot\) The authors consider a one-parameter family of curves \(\gamma_t\in \mathrm{Emb}(\mathbb S^1,M)\) solutions of
\[\partial_t\gamma_t(u)=\omega_t(u)N_{\gamma_t}(u),\tag{1}\]
where \(\omega_t(u)\) is an explicit expression given in terms of the partial derivatives of \(F\) with respect to some local coordinates on \(M\), \(N_{\gamma_t}(u)\), and on the norm of \(\dot{\gamma_t}\). Then, they state that if \(\gamma_t\) is the solution of (1) with initial condition \(\gamma_0\), then there is a unique \(\mathrm{Emb}(\mathbb S^1,M)\)-valued continuous map \(\varphi\) on an open neighborhood of \(\{0\}\times\mathrm{Emb}(\mathbb S^1,M)\) such that \(\varphi(t,\gamma_0)=\varphi_t(\gamma_0)=\gamma_t\). Furthermore, \(\varphi_t(\gamma\circ\theta)=\varphi_t(\gamma)\circ\theta\) for \(\theta\in Diff(\mathbb S^1)\), \(\frac{d}{dt}L(\varphi_t(\gamma))\le 0\) for \(\gamma\in \mathrm{Emb}(\mathbb S^1,M)\) where \(L\) is the Finsler length functional on \(\mathrm{Emb}(\mathbb S^1,M)\) defined as \(L(\gamma)=\displaystyle\int_0^1F(\gamma(t),\dot{\gamma_t}(u))du\), and if \(l_\gamma=\displaystyle\lim_{t\to\tau_\gamma}L(\varphi_t(\gamma))>0\), then \(\tau_\gamma=\infty\) (Theorem 2.1). Also, they state that every reversible Finsler two-sphere \((\mathbb S^2,F)\) has at least three explicit geometrically distinct simple closed geodesics (Theorem 1.3).
Reviewer: Mohammed El Aïdi (Bogotá)Gaussian free fields and Riemannian rigidityhttps://zbmath.org/1496.530612022-11-17T18:59:28.764376Z"Nguyen Viet Dang"https://zbmath.org/authors/?q=ai:nguyen-viet-dang.Summary: On a compact Riemannian manifold \((M,g)\) of dimension \(d\leqslant 4\), we present a rigorous construction of the renormalized partition function \(Z_g(\lambda)\) of a massive Gaussian free field where we explicitly determine the local counterterms using microlocal methods. Then we show that \(Z_g(\lambda)\) determines the Laplace spectrum of \((M,g)\) and hence imposes some strong geometric constraints on the Riemannian structure of \((M,g)\). From this observation, using classical results in Riemannian geometry, we illustrate how the partition function allows us to probe the Riemannian structure of the underlying manifold \((M,g)\).On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz-Minkowski spacetime \(\mathbb{L}^m\)https://zbmath.org/1496.530732022-11-17T18:59:28.764376Z"Palomo, Francisco J."https://zbmath.org/authors/?q=ai:palomo.francisco-j"Romero, Alfonso"https://zbmath.org/authors/?q=ai:romero.alfonsoAccording to the Reilly inequality [\textit{R. C. Reilly}, Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)] for a compact submanifold in the Euclidean space \(\mathbf{E}^m\), it was known that the first non-trivial eigenvalue \(\lambda_1\) of the Laplacian of the induced metric on \(M\) satisfies the following inequality:
\[\lambda_1\geq n\frac{\int_{M}||\mathbf{H}||^{2}dV}{\text{Vol}(M)}.\] The equality holds if and only if \(M\) lies minimally in some hypersphere in \(\mathbf{E}^m\). \newline In this paper, the authors provide an alternate inequality for any compact space-like submanifold \(M\) of the Lorentz-Minkowski spacetime \(\mathbf{L}^m\). Their results show that for each unit time-like vector \(a\in \mathbf{L}^m\), the first nontrivial eigenvalue \(\lambda_1\) satisfies \[\lambda_1\leq n\frac{\int_{M}(||\mathbf{H}||^2+\langle\mathbf{H},a\rangle^2)dV}{\text{Vol}(M)+1/n\int_{M}||a^{T}||^2dV},\] where \(a^T\) is the orthogonal projection of the vector \(a\) on \(T_pM\). Several interesting results are provided along with a clear geometric meaning. By interpreting a compact submanifold \(M\) of a Euclidean space as a compact space-like submanifold of Lorentz-Minkowski spacetime through a space-like hyperplane, the original Reilly inequality is proved also.
Reviewer: Yun Myung Oh (Berrien Springs)Every symplectic manifold is a (linear) coadjoint orbithttps://zbmath.org/1496.530812022-11-17T18:59:28.764376Z"Donato, Paul"https://zbmath.org/authors/?q=ai:donato.paul"Iglesias-Zemmour, Patrick"https://zbmath.org/authors/?q=ai:iglesias-zemmour.patrickRecall that the Kirillov-Kostant-Souriau theorem assures that a symplectic manifold that is homogeneous under the action of a Lie group is isomorphic, up to a covering, to a possibly affine coadjoint orbit.
The authors generalize this result. They prove that every symplectic manifold is a coadjoint orbit of the diffeological group of automorphisms of its integration bundle. The integration bundle is a principal fiber bundle over the manifold, with group the torus of periods of the symplectic form, quotient of the real line by the group of periods, i.e., the integrals of the two-form on every two-cycle.
Reviewer: Daniele Angella (Firenze)Volume properties and rigidity on self-expanders of mean curvature flowhttps://zbmath.org/1496.530952022-11-17T18:59:28.764376Z"Ancari, Saul"https://zbmath.org/authors/?q=ai:ancari.saul"Cheng, Xu"https://zbmath.org/authors/?q=ai:cheng.xuSummary: In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the \(L\)-stability operator and discuss the \(L\)-stability of some special self-expanders. Besides, we prove that the surfaces \(\Gamma \times \mathbb{R}\) with the product metric are the only complete self-expander surfaces immersed in \(\mathbb{R}^3\) with constant scalar curvature, where \(\Gamma\) is a complete self-expander curve (properly) immersed in \(\mathbb{R}^2\).Elliptic gradient estimates for a nonlinear \(f\)-heat equation on weighted manifolds with evolving metrics and potentialshttps://zbmath.org/1496.531002022-11-17T18:59:28.764376Z"Abolarinwa, Abimbola"https://zbmath.org/authors/?q=ai:abolarinwa.abimbola"Taheri, Ali"https://zbmath.org/authors/?q=ai:taheri.ali|taheri.ali-karimiSummary: We develop local elliptic gradient estimates for a basic nonlinear \(f\)-heat equation with a logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in the context of weighted manifolds, where the metric and potential evolve under a Perelman-Ricci type flow. For the heat bounds use is made of entropy monotonicity arguments and ultracontractivity estimates with the bounds expressed in terms of the optimal constant in the logarithmic Sobolev inequality. Some interesting consequences of these estimates are presented and discussed.On the regularity of Ricci flows coming out of metric spaceshttps://zbmath.org/1496.531012022-11-17T18:59:28.764376Z"Deruelle, Alix"https://zbmath.org/authors/?q=ai:deruelle.alix"Schulze, Felix"https://zbmath.org/authors/?q=ai:schulze.felix"Simon, Miles"https://zbmath.org/authors/?q=ai:simon.milesThe authors investigate the following problem: consider a possibly incomplete Ricci flow \((M,g(t))_{t \in (0,T)}\) satisfying
\[ |\mathrm{Rm}|\leq c_0/t \tag{1} \]
and whose associated metric spaces \((M,d_{g(t)})\) Gromov-Hausdorff converge to a metric space \((X,d_X)\) as \(t \searrow 0\). What additional assumptions on \((X,d_X)\) and \((M,g(t))_{t \in (0,T)}\) guarentee that \(g(t)\) converges locally smoothly (or continuously) to a smooth (or continous) metric as \(t \searrow 0\)?
The authors give a positive answer under the additional assumption that for all \(t \in (0,T)\) \[ \mathrm{Ric}_{g(t)}\geq -1, \tag{2}\] \(B_{g(t)}(x_0,1) \Subset M\) and that \((X,d_X)\) is \emph{smoothly (or continously) \(n\)-Riemannian}. One says that \((X,d_X)\) is \emph{smoothly (respectively continously) \(n\)-Riemannian} if for any \(x_0 \in X\) there are \(0<\tilde r<r\) with \(\tilde r<r/5\) and points \(a_1,\ldots,a_n\in B_{d_X}(x_0,r)\) such that the map \[F(x) = (d_X(a_1,x),\ldots,d_X(a_n,x)),\quad x \in B_{d_X}(x_0,r)\] is an \((1+\varepsilon_0)\)-bilipschitz homeomorphism on \(B_{d_X}(x_0,5\tilde r)\), and the pushed forward of \(d_X\) via \(F\) is on \(\mathbb{B}_{4\tilde r}(F(x_0))\Subset F(B_{d_X}(x_0,5\tilde r))\) induced by a smooth (respectively continous) Riemannian metric. The smooth definition corresponds to \((X,d_X)\) being locally isometric to smooth \(n\)-Riemannian manifolds.
Before stating the main results let us recall a bit of context. When all \(g(t)\) are complete and satisfy (1) and (2), \textit{M. Simon} and \textit{P. M. Topping} [Geom. Topol. 25, No. 2, 913--948 (2021; Zbl 1470.53083)] show that \(d_{g(t)} \to d_0\) a metric on \(M\) as \(t \searrow 0\), and that \((M,d_0)\) is isometric to \((X,d_X)\). Without the completeness assumption, a localized version of this holds assuming that \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\). Then \(X:=\cap_{s \in (0,T)} B_{g(s)}(x_0,1/2)\) is non empty and is endowed with a well defined limiting metric \(d_0=\lim d_{g(t)}\) as \(t \searrow 0\). Moreover \(B_{g(t)}(x_0,r) \Subset \mathcal{X} \subset X\) for all \(r \leq R(c_0,n)\) and \(t \leq S(c_0,n)\), where \(\mathcal{X}\) is the connected component of \(X\) containing \(x_0\), and the topology of \(B_{d_0}(x_0,r)\) induced by \(d_0\) agrees with that of \(M\).
The main results of the authors are the following.
Theorem 1.6 : Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is smoothly \(n\)-Riemannian for some \(r<R(c_0,n)\). Then there exists a smooth Riemannian metric \(g_0\) on \(B_{d_0}(x_0,s)\) for some \(s<r\) such that \(g(t)\) extends to a smooth solution \((B_{d_0}(x_0,s), g(t))_{t \in [0,T)}\) by defining \(g(0)=g_0\).
Theorem 1.7: Let \((M,g(t))_{t \in (0,T)}\) be a Ricci flow satisfying (1) and (2), assume \(B_{g(t)}(x_0,1) \Subset M\) for all \(t \in (0,T)\) and let \((X,d_0)\) be the limit as above. Assume further that \((B_{d_0}(x_0,r),d_0)\) is continuously \(n\)-Riemannian for some \(r<R(c_0,n)\). Then for any stricty monotone sequence \(t_i \searrow 0\), there exists \(v>0\) and a continuous Riemannian metric \(\tilde g_0\) defined on \(\mathbb{B}_v(p) \subset \mathbb{R}^n\) and a family of smooth diffeomorphisms \(Z_i : B_{d_0}(x_0,2v) \to \mathbb{R}^n\) such that \((Z_i)_\ast(g(t_i))\) converges in the \(C^0\)-sense to \(\tilde g_0\) as \(t_i \searrow\) on \(\mathbb{B}_v(p)\).
Reviewer: Laurent Bessières (Bordeaux)Persistent Laplacians: properties, algorithms and implicationshttps://zbmath.org/1496.550072022-11-17T18:59:28.764376Z"Mémoli, Facundo"https://zbmath.org/authors/?q=ai:memoli.facundo"Wan, Zhengchao"https://zbmath.org/authors/?q=ai:wan.zhengchao"Wang, Yusu"https://zbmath.org/authors/?q=ai:wang.yusuThis paper studies the \textit{persistent Laplacian}, i.e. an extension of the ordinary combinatorial Laplacian to sequences of simplicial complexes. Given two pairs of simplicial complexes \(K\) and \(L\), with \(K \hookrightarrow L\), the authors introduce an effective algorithm for finding a matrix representation of the \(q\)th persistent Laplacian \(\Delta_q^{K, L}\). Among other things, this enables the calculation of the \(q\)th persistent Betti number of the inclusion \(K \hookrightarrow L\). Next to these results, the authors also prove properties of the spectrum of \(\Delta_q^{K, L}\), as well as a link between its nullity and persistent Betti numbers. Furthermore, the persistent Laplacian is linked to the notion of effective resistance in graphs. This paper hence makes the previously-introduced concept of a persistent Laplacian effectively computable, providing both a highly-relevant formalisation as well as a contextualisation.
Reviewer: Bastian Rieck (Bern)The Atiyah-Patodi-Singer rho invariant and signatures of linkshttps://zbmath.org/1496.570102022-11-17T18:59:28.764376Z"Toffoli, Enrico"https://zbmath.org/authors/?q=ai:toffoli.enricoMultivariable signatures of links, defined by \textit{D. Cimasoni} and \textit{V. Florens} [Trans. Am. Math. Soc. 360, No. 3, 1223--1264 (2008; Zbl 1132.57004)] using generalized Seifert surfaces, were given a description as twisted signatures of four manifolds and also employed as concordance invariants. However, a description of them as Atiyah-Patodi-Singer rho invariants was not investigated thoroughly.
In the first part of the paper, the author fills this gap by proving a cut-and-paste formula for the Atiyah-Patodi-Singer rho invariant which allows him to manipulate manifolds in a convenient way. He gives a simple proof which does not involve rho invariants of manifolds with boundary, and which is based on Wall's non-additivity for the signature instead.
The second part of the paper is made up of applications of the main Theorem in the context of link theory. The author gives a description of the multivariable signature of a link \(L\) as the rho invariant of some closed three-manifold \(Y_L\) intrinsically associated with \(L\). Then, he studies the rho invariant of the manifolds obtained by Dehn surgery on \(L\) along integer and rational framings. Inspired by the results of \textit{A. J. Casson} and \textit{C. McA. Gordon} [in: Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., Vol. 32, Part 2, 39--53 (1978; Zbl 0394.57008)] and Cimasoni and Florens [loc. cit.], he gives formulas expressing this value as a sum of the multivariable signature of \(L\) and some easy-to-compute extra terms.
Reviewer: Leila Ben Abdelghani (Monastir)Equivariant spectral triples for homogeneous spaces of the compact quantum group \(U_q(2)\)https://zbmath.org/1496.580012022-11-17T18:59:28.764376Z"Guin, Satyajit"https://zbmath.org/authors/?q=ai:guin.satyajit"Saurabh, Bipul"https://zbmath.org/authors/?q=ai:saurabh.bipulSummary: In this article, we study homogeneous spaces \(U_q(2)/_\phi\mathbb{T}\) and \(U_q(2)/_\psi\mathbb{T}\) of the compact quantum group \(U_q(2)\), \(q\in\mathbb{C}\setminus \{0\}\). The homogeneous space \(U_q(2)/_\phi\mathbb{T}\) is shown to be the braided quantum group \(SU_q(2)\). The homogeneous space \(U_q(2)/_\psi\mathbb{T}\) is established as a universal \(C^\ast\)-algebra given by a finite set of generators and relations. Its \(\mathcal{K}\)-groups are computed and two families of finitely summable odd spectral triples, one is \(U_q(2)\)-equivariant and the other is \(\mathbb{T}^2\)-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the \(\mathcal{K}\)-homology group \(K^1(C(U_q(2)/_\psi\mathbb{T}))\).Short survey on the existence of slices for the space of Riemannian metricshttps://zbmath.org/1496.580022022-11-17T18:59:28.764376Z"Corro, Diego"https://zbmath.org/authors/?q=ai:corro.diego"Kordaß, Jan-Bernhard"https://zbmath.org/authors/?q=ai:kordass.jan-bernhardThe article in question gives a survey about slice theorems and an alternative proof of Ebin's slice theorem. The latter roughly states, that any Riemannian metric \(g\) on a given manifold \(M\) admits an open neighbourhood diffeomorphic to \(S_g\times(\mathrm{Diff(M)}/{\mathrm{Diff}(M)_g})\), where \(\mathrm{Diff}(M)\) is the group of self-diffeomorphisms of \(M\), \(\mathrm{Diff}(M)_g\) is the subgroup of isometries of \(g\) and \(S_g\) is a certain submanifold of the space of all Riemannian metrics called the slice. In contrast to the original proof of \textit{D. G. Ebin} [Proc. Sympos. Pure Math. 15, 11--40 (1970; Zbl 0205.53702)], the authors avoid technical work done in the context of Sobolev spaces. They are also able to show the existence of an equivariant tubular neighbourhood of the orbit \(\mathrm{Diff}(M)\cdot g\) which is homeomorphic to \(\mathrm{Diff}(M)\times_{\mathrm{Diff}(M)_g} S_g\).
For the entire collection see [Zbl 1495.53005].
Reviewer: Georg Frenck (Augsburg)A variational principle for Kaluza-Klein types theorieshttps://zbmath.org/1496.580032022-11-17T18:59:28.764376Z"Hélein, Frédéric"https://zbmath.org/authors/?q=ai:helein.fredericSummary: For any positive integer \(n\) and any Lie group \(\mathfrak{G}\), given a definite symmetric bilinear form on \(\mathbb{R}^n\) and an Ad-invariant scalar product on the Lie algebra of \(\mathfrak{G} \), we construct a variational problem on fields defined on an arbitrary oriented \((n + \dim\mathfrak{G})\)-dimensional manifold \(\mathcal{Y}\). We show that, if \(\mathfrak{G}\) is compact and simply connected, any global solution of the Euler-Lagrange equations leads, through a spontaneous symmetry breaking, to identify \(\mathcal{Y}\) with the total space of a principal bundle over an n-dimensional manifold \(\mathcal{X} \). Moreover \(\mathcal{X}\) is then endowed with a (pseudo-)Riemannian metric and a connection which are solutions of the Einstein-Yang-Mills system of equations with a cosmological constant.Relaxed energies, defect measures, and minimal currentshttps://zbmath.org/1496.580042022-11-17T18:59:28.764376Z"Lin, Fang-Hua"https://zbmath.org/authors/?q=ai:lin.fang-huaA natural existence question for a continuous harmonic map with a suitably given Dirichlet boundary value or in a given homotopic class (a problem posed by R. Schoen) remains open. The author briefly describes several earlier studies concerning energy minimizing harmonic maps, and maps that minimize the so-called relaxed energy from \(\mathbb{R}^3\) into \(S^2\). Of particular interest is the partial regularity and properties of possible singularities of such maps. A sketch proof of a formula conjectured by \textit{H. Brezis} and \textit{P. Mironescu} [Sobolev maps to the circle. From the perspective of analysis, geometry, and topology. New York, NY: Birkhäuser (2021; Zbl 07332819)] is provided, concerning the relaxed \(k\)-energy for Sobolev maps from \(\mathbb{R}^n\) to \(S^k\), for \(k>1\).
For the entire collection see [Zbl 1491.46003].
Reviewer: Vladimir Balan (Bucureşti)A Cheeger-Müller theorem for manifolds with wedge singularitieshttps://zbmath.org/1496.580052022-11-17T18:59:28.764376Z"Albin, Pierre"https://zbmath.org/authors/?q=ai:albin.pierre"Rochon, Frédéric"https://zbmath.org/authors/?q=ai:rochon.frederic"Sher, David"https://zbmath.org/authors/?q=ai:sher.david-aThe authors study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, the authors give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd-dimensional, this is used to obtain a Cheeger-Mueller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.
Reviewer: Shu-Yu Hsu (Chiayi)An index formula for groups of isometric linear canonical transformationshttps://zbmath.org/1496.580062022-11-17T18:59:28.764376Z"Savin, Anton"https://zbmath.org/authors/?q=ai:savin.anton-yu"Schrohe, Elmar"https://zbmath.org/authors/?q=ai:schrohe.elmarAuthors' abstract: We define a representation of the unitary group \(U(n)\) by metaplectic operators acting on \( L^2(\mathbb{R}^n) \) and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.
Reviewer: Jan Kurek (Lublin)The derivatives of the heat kernel on complete manifoldshttps://zbmath.org/1496.580072022-11-17T18:59:28.764376Z"Fotiadis, Anestis"https://zbmath.org/authors/?q=ai:fotiadis.anestisThe author uses a new iteration argument to obtain the estimates for the time derivatives of the heat kernel on complete non-compact manifolds. Then the author applies these estimates to study the \(L^p\)-boundedness of the Littlewood-Paley-Stein operators on a class of locally symmetric spaces.
Reviewer: Shu-Yu Hsu (Chiayi)Gradient estimates of a nonlinear elliptic equation for the \(V\)-Laplacian on noncompact Riemannian manifoldshttps://zbmath.org/1496.580082022-11-17T18:59:28.764376Z"Yihua, Deng"https://zbmath.org/authors/?q=ai:yihua.dengSummary: In this paper, we consider gradient estimates for positive solutions to the following equation
\[
\triangle_V u+au^p\log u=0
\]
on complete noncompact Riemannian manifold with \(k\)-dimensional Bakry-Émery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of \(|\nabla u|\) with respect to the lower bound of the Bakry-Émery Ricci curvature.Spectral geometry on manifolds with fibered boundary metrics. I: Low energy resolventhttps://zbmath.org/1496.580092022-11-17T18:59:28.764376Z"Grieser, Daniel"https://zbmath.org/authors/?q=ai:grieser.daniel"Talebi, Mohammad"https://zbmath.org/authors/?q=ai:talebi.mohammad-sadegh"Vertman, Boris"https://zbmath.org/authors/?q=ai:vertman.borisIn the paper under review, the authors investigate the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric, and they determine the precise asymptotic behavior of the resolvent as a fibered boundary pseudodifferential operator when the resolvent parameter tends to zero.
This work constitutes a generalization of previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibers are points.
The novel aspect in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise harmonic and on its orthogonal complement. In order for the authors to treat this, they introduce an appropriate `split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker.
Reviewer: Themistocles M. Rassias (Athína)Resolvent estimates on asymptotically cylindrical manifolds and on the half linehttps://zbmath.org/1496.580102022-11-17T18:59:28.764376Z"Christiansen, Tanya J."https://zbmath.org/authors/?q=ai:christiansen.tanya-j"Datchev, Kiril"https://zbmath.org/authors/?q=ai:datchev.kiril-rIn this article the authors study the spectral and scattering theory for a class of asympotically cylindrical manifolds with sufficiently mild geodesic trapping. One example of such a manifold is the cigar-shaped warped product (\(\mathbb{R}^d\), \(g_0\)), \(d \ge 2\), with metric \(g_0 = dr^2 + F(r) dS\), where \(r\) is the radial variable, \(dS\) is the usual metric on the unit sphere, and \(F(r) = r^2\) near \(r = 0\), while \(F'\) is compactly supported on some interval \([0,R]\) and positive on \((0,R)\). Another example is a convex cocompact hyperbolic surface \((X,g_H)\) for which there is a compact set \(N \subseteq X\) such that
\[
X \setminus N = (0,\infty)_r \times Y_y, \quad g_H\rvert_{X \setminus N} = dr^2 + \cosh^2 r dy^2,
\]
where \(Y\) is a disjoint union of \(k \ge 1\) geodesic circles. Indeed, in the first example, the only trapped geodesics are the circular ones on the cylindrical end (and this is the smallest amount of trapping a manifold with a cylindrical end can have.)
Suppose that trapping is suitably mild, in the sense that, in the presence of complex absorption, the resolvent is bounded polynomially in the spectral parameter \(z\) as \(\text{Re}\, z \to \infty\). Then the authors show that the number of embedded resonances and eigenvalues is finite, and that the cutoff resolvent (without complex absoprtion) is uniformly bounded as \(\text{Re}\, z \to \infty\). This bound is sharp in the setting of the first example described above.
Along the way to their main result, the authors also prove some resolvent estimates for repulsive potentials on the half-line.
Reviewer: Jacob Shapiro (Dayton)Restriction of Laplace-Beltrami eigenfunctions to arbitrary sets on manifoldshttps://zbmath.org/1496.580112022-11-17T18:59:28.764376Z"Eswarathasan, Suresh"https://zbmath.org/authors/?q=ai:eswarathasan.suresh"Pramanik, Malabika"https://zbmath.org/authors/?q=ai:pramanik.malabikaAuthors' abstract: Given a compact Riemannian manifold \((M, g)\) without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets \(\Gamma\) of \(M\). The sets \(\Gamma\) that we consider are Borel measurable, Lebesguenull but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on \(\Gamma\). For large Lebesgue exponents \(p\), these estimates provide a natural generalization of \(L^p\) bounds for eigenfunctions restricted to submanifolds, previously obtained in [\textit{N. Burq} et al., Duke Math. J. 138, No. 3, 445--486 (2007; Zbl 1131.35053); \textit{L. Hörmander}, Acta Math. 121, 193--218 (1968; Zbl 0164.13201); Ark. Mat. 11, 1--11 (1973; Zbl 0254.42010); \textit{C. D. Sogge}, J. Funct. Anal. 77, No. 1, 123--138 (1988; Zbl 0641.46011)]. Under an additional measure-theoretic assumption on \(\Gamma\), the estimates are shown to be sharp in this range. As evidence of the genericity of the sharp estimates, we provide a large family of random, Cantor-type sets that are not submanifolds, where the above-mentioned sharp bounds hold almost surely.
Reviewer: Mohammed El Aïdi (Bogotá)Positive entropy using Hecke operators at a single placehttps://zbmath.org/1496.580122022-11-17T18:59:28.764376Z"Shem-Tov, Zvi"https://zbmath.org/authors/?q=ai:shemtov.zviSummary: We prove the following statement: let \(X=\mathrm{SL}_n(\mathbb{Z})\backslash\mathrm{SL}_n(\mathbb{R})\) and consider the standard action of the diagonal group \(A<\mathrm{SL}_n(\mathbb{R})\) on it. Let \(\mu\) be an \(A\)-invariant probability measure on \(X\), which is a limit
\[
\mu=\lambda\lim\limits_i|\phi_i|^2dx,
\]
where \(\phi_i\) are normalized eigenfunctions of the Hecke algebra at some fixed place \(p\) and \(\lambda>0\) is some positive constant. Then any regular element \(a\in A\) acts on \(\mu\) with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over \(\mathbb{Q}\) and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of \textit{S. Brooks} and \textit{E. Lindenstrauss} [Invent. Math. 198, No. 1, 219--259 (2014; Zbl 1343.58016)].An averaging principle for stochastic flows and convergence of non-symmetric Dirichlet formshttps://zbmath.org/1496.600942022-11-17T18:59:28.764376Z"Barret, Florent"https://zbmath.org/authors/?q=ai:barret.florent"Raimond, Olivier"https://zbmath.org/authors/?q=ai:raimond.olivierLet \(X^\kappa\) be a diffusion process on a Riemannian manifold \(M\) with generator \(A^\kappa=A+\kappa V\), where \(A\) is a second order differential operator, \(V\) is a vector field on \(M\) and \(\kappa\) is a large positive parameter. After an appropriate time change, \(X^\kappa\) is a random perturbation of the dynamical system \(\frac{dx_t}{dt}=V(x_t)\). Suppose that one is able to construct a certain metric space \(\tilde{M}\) and a continuous map \(\pi: M\rightarrow \tilde{M}\). Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco convergence, the paper under review mainly obtains that under suitable conditions, \(\tilde{X}^\kappa:=\pi(X^\kappa)\) converges in law towards a diffusion process \(\tilde{X}\) on \(\tilde{M}\) as \(\kappa\rightarrow \infty\). This result permits higher dimensions than \(d=2\), and the case of dimension \(2\) was paid particular attention to in the book [\textit{M. I. Freidlin} and \textit{A. D. Wentzell}, Random perturbations of dynamical systems. Translated from the Russian by J Szücs. 3rd ed. Berlin: Springer (2012; Zbl 1267.60004)]. Furthermore, this result is also applied to a system of \(n\) particles in a turbulent fluid with a shear flow generated by \(V\).
Reviewer: Liping Li (Beijing)Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolationhttps://zbmath.org/1496.652032022-11-17T18:59:28.764376Z"Cheng, Xiuyuan"https://zbmath.org/authors/?q=ai:cheng.xiuyuan"Wu, Nan"https://zbmath.org/authors/?q=ai:wu.nanSummary: We study the spectral convergence of graph Laplacians to the Laplace-Beltrami operator when the kernelized graph affinity matrix is constructed from \(N\) random samples on a \(d\)-dimensional manifold in an ambient Euclidean space. By analyzing Dirichlet form convergence and constructing candidate approximate eigenfunctions via convolution with manifold heat kernel, we prove eigen-convergence with rates as \(N\) increases. The best eigenvalue convergence rate is \(N^{-1/(d/2 +2)}\) (when the kernel bandwidth parameter \(\epsilon \sim (\log N/N)^{1/(d/2+2)}\)) and the best eigenvector 2-norm convergence rate is \(N^{-1/(d/2+3)}\) (when \(\epsilon \sim (\log N / N )^{1/(d/2+3)}\)). These rates hold up to a \(\log N\)-factor for finitely many low-lying eigenvalues of both un-normalized and normalized graph Laplacians. When data density is non-uniform, we prove the same rates for the density-corrected graph Laplacian, and we also establish new operator point-wise convergence rate and Dirichlet form convergence rate as intermediate results. Numerical results are provided to support the theory.Error correction of the continuous-variable quantum hybrid computation on two-node cluster states: limit of squeezinghttps://zbmath.org/1496.810442022-11-17T18:59:28.764376Z"Korolev, S. B."https://zbmath.org/authors/?q=ai:korolev.s-b"Golubeva, T. Yu."https://zbmath.org/authors/?q=ai:golubeva.t-yuSummary: In this paper, we investigate the error correction of universal Gaussian transformations obtained in the process of continuous-variable quantum computations. We have tried to bring our theoretical studies closer to the actual picture in the experiment. When investigating the error correction procedure, we have considered that both the resource GKP state itself and the entanglement transformation are imperfect. In reality, the GKP state has a finite width associated with the finite degree of squeezing, and the entanglement transformation is performed with error. We have considered a hybrid scheme to implement the universal Gaussian transformations. In this scheme, the transformations are realized through computations on the cluster state, supplemented by linear optical operation. This scheme gives the smallest error in the implementation of universal Gaussian transformations. The use of such a scheme made it possible to reduce the oscillator squeezing threshold required for the implementing of fault-tolerant quantum computation schemes close to reality to \(-19.25\) dB.The noncommutative space of light-like worldlineshttps://zbmath.org/1496.810692022-11-17T18:59:28.764376Z"Ballesteros, Angel"https://zbmath.org/authors/?q=ai:ballesteros.angel"Gutierrez-Sagredo, Ivan"https://zbmath.org/authors/?q=ai:gutierrez-sagredo.ivan"Herranz, Francisco J."https://zbmath.org/authors/?q=ai:herranz.francisco-joseSummary: The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) \(\kappa\)-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter \(\kappa^{-1}\). Moreover, it is shown that the usual time-like \(\kappa\)-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like \(\kappa\)-deformation.Special cosmological models derived from the semiclassical Einstein equation on flat FLRW space-timeshttps://zbmath.org/1496.830022022-11-17T18:59:28.764376Z"Gottschalk, Hanno"https://zbmath.org/authors/?q=ai:gottschalk.hanno"Rothe, Nicolai R."https://zbmath.org/authors/?q=ai:rothe.nicolai-r"Siemssen, Daniel"https://zbmath.org/authors/?q=ai:siemssen.danielThe Einstein equations and multipole moments at null infinityhttps://zbmath.org/1496.830032022-11-17T18:59:28.764376Z"Tafel, J."https://zbmath.org/authors/?q=ai:tafel.jacekGeneralized geodesic deviation in de Sitter spacetimehttps://zbmath.org/1496.830142022-11-17T18:59:28.764376Z"Waldstein, Isaac Raj"https://zbmath.org/authors/?q=ai:waldstein.isaac-raj"Brown, J. David"https://zbmath.org/authors/?q=ai:brown.j-davidThe Penrose property with a cosmological constanthttps://zbmath.org/1496.830312022-11-17T18:59:28.764376Z"Cameron, Peter"https://zbmath.org/authors/?q=ai:cameron.peter-jLaplacian on fuzzy de Sitter spacehttps://zbmath.org/1496.830332022-11-17T18:59:28.764376Z"Brkić, Bojana"https://zbmath.org/authors/?q=ai:brkic.bojana"Burić, Maja"https://zbmath.org/authors/?q=ai:buric.maja"Latas, Duško"https://zbmath.org/authors/?q=ai:latas.duskoInteraction of inhomogeneous axions with magnetic fields in the early universehttps://zbmath.org/1496.830382022-11-17T18:59:28.764376Z"Dvornikov, Maxim"https://zbmath.org/authors/?q=ai:dvornikov.maximSummary: We study the system of interacting axions and magnetic fields in the early universe after the quantum chromodynamics phase transition, when axions acquire masses. Both axions and magnetic fields are supposed to be spatially inhomogeneous. We derive the equations for the spatial spectra of these fields, which depend on conformal time. In case of the magnetic field, we deal with the spectra of the energy density and the magnetic helicity density. The evolution equations are obtained in the closed form within the mean field approximation. We choose the parameters of the system and the initial condition which correspond to realistic primordial magnetic fields and axions. The system of equations for the spectra is solved numerically. We compare the cases of inhomogeneous and homogeneous axions. The evolution of the magnetic field in these cases is different only within small time intervals. Generally, magnetic fields are driven mainly by the magnetic diffusion. We find that the magnetic field instability takes place for the amplified initial wavefunction of the homogeneous axion. This instability is suppressed if we account for the inhomogeneity of the axion.Observational constraints on inflection point quintessence with a cubic potentialhttps://zbmath.org/1496.830482022-11-17T18:59:28.764376Z"Storm, S. David"https://zbmath.org/authors/?q=ai:storm.s-david"Scherrer, Robert J."https://zbmath.org/authors/?q=ai:scherrer.robert-jSummary: We examine the simplest inflection point quintessence model, with a potential given by \(V(\phi) = V_0 + V_3\phi^3\). This model can produce either asymptotic de Sitter expansion or transient acceleration, and we show that it does not correspond to either pure freezing or thawing behavior. We derive observational constraints on the initial value of the scalar field, \(\phi_i\), and \(V_3/V_0\) and find that small values of either \(\phi_i\) or \(V_3/V_0\) are favored. While most of the observationally-allowed parameter space yields asymptotic de Sitter evolution, there is a small region, corresponding to large \(V_3/V_0\) and small \(\phi_i\), for which the current accelerated expansion is transient. The latter behavior is potentially consistent with a cyclic universe.Nilpotent singularities and chaos: tritrophic food chainshttps://zbmath.org/1496.920892022-11-17T18:59:28.764376Z"Drubi, Fátima"https://zbmath.org/authors/?q=ai:drubi.fatima"Ibáñez, Santiago"https://zbmath.org/authors/?q=ai:ibanez.santiago"Pilarczyk, Paweł"https://zbmath.org/authors/?q=ai:pilarczyk.pawelSummary: Local bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding this method and describe the main steps to illustrate numerically the chaotic dynamics emerging near these nilpotent singularities. This is a general-purpose method and we hope it can be applied to a huge range of models.