Recent zbMATH articles in MSC 58https://zbmath.org/atom/cc/582023-01-20T17:58:23.823708ZWerkzeugJets and differential linear logichttps://zbmath.org/1500.030182023-01-20T17:58:23.823708Z"Wallbridge, James"https://zbmath.org/authors/?q=ai:wallbridge.jamesThis paper studies differential linear logic [\textit{T. Ehrhard}, Math. Struct. Comput. Sci. 28, No. 7, 995--1060 (2018; Zbl 1456.03097)] through the lens of the category of vector bundles over a smooth manifold. It is established that a number of categories arising from the category of vector bundles are models of intuitionistic differential linear logic. This is part of a larger project aimed at understanding the interaction between differential programming based on differential \(\lambda\)-calculus [\textit{T. Ehrhard} and \textit{L. Regnier}, Theor. Comput. Sci. 309, No. 1--3, 1--41 (2003; Zbl 1070.68020)] and differential linear logic with a view towards extending these concepts to a language of non-linear partial differential equations. Since morphisms come from proofs in differential linear logic, and proofs are identified with programs in differential \(\lambda\)-calculus via the Curry-Howard correspondence, the denotational semantics provide tools for differentiable programming.
There exist a number of approaches to the categorical semantics of differential linear logic in the literature, including Köthe sequence spaces [\textit{T. Ehrhard}, Math. Struct. Comput. Sci. 12, No. 5, 579--623 (2002; Zbl 1025.03066)], finiteness spaces [\textit{T. Ehrhard}, Math. Struct. Comput. Sci. 15, No. 4, 615--646 (2005; Zbl 1084.03048)], convenient vector spaces [\textit{R. Blute} et al., Cah. Topol. Géom. Différ. Catég. 53, No. 3, 211--232 (2012; Zbl 1281.46061)] and vector spaces themselves [\textit{J. Clift} and \textit{D. Murfet}, Math. Struct. Comput. Sci. 30, No. 4, 416--457 (2020; Zbl 1495.03075)]. The author's approach begins by considering a smooth generalization of [loc. cit.] where the underlying objects are vector spaces parametrized by a fixed base manifold \(M\). To a formula \(A\) in differential linear logic, the sheaf \(\mathcal{E}\) of sections of a vector bundle \(E\) on \(M\) is associated. When \(E\) is the trivial line bundle, the associated denotation is simply the sheaf \(\mathcal{C}_{M}^{\infty}\) of smooth functions on \(M\).
It is shown that there are two natural comonads on the category of vector bundles to model the exponential modality of linear logic. First, there is the \textit{jet comonad} \(!_{j}\) [\textit{M. Marvan}, in: Differential geometry and its applications. Proceedings of the conference on differential geometry and its applications, August 24--30, 1986, Brno, Czechoslovakia. Communications. Brno: J. E. Purkyně University. 235--244 (1987; Zbl 0629.58033)], which sends a sheaf \(\mathcal{E}\) to the sheaf \(!_{j}\mathcal{E}\) of infinite jets of local sections of \(E\). The idea of a syntactic Taylor expansion in linear logic and \(\lambda\)-calculus through the exponential connective [\textit{T. Ehrhard} and \textit{L. Regnier}, Theor. Comput. Sci. 403, No. 2--3, 347--372 (2008; Zbl 1154.68354)] is explicitly present in the semantics of vector bundles. Working in the general setting of infinite jets, as opposed to \(r\)-jets for a fixed \(r\in\mathbb{N}\), forces one to work in the enlarged category of pro-ind vector bundles [\textit{B. Güneysu} and \textit{M. J. Pflaum}, SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 003, 44 p. (2017; Zbl 1367.58002)]. The objects in this category are (co)filtered objects in the category of vector bundles on \(M\). The jet construction makes direct contact with the theory of differential operators and linear partial differential equations, which enables one to understand these concepts within the setting of differential linear logic. The Kleisli category for the jet comonad is the category of convenient vector bundles \(\mathcal{E}\) on \(M\) and whose morphisms \(!_{j}\mathcal{E}\rightarrow\mathcal{E}\) are linear differential operators. The category is equivalent to the category of infinitely prolongated linear partial differential equations on \(M\) as its manifold of independent variables. It is shown that the category of convenient \(\mathcal{C}_{M}^{\infty}\)-modules with the jet comonad is a symmetric monoidal storage category in the sense of [\textit{R. Blute} et al., Cah. Topol. Géom. Différ. Catég. 53, No. 3, 211--232 (2012; Zbl 1281.46061)].
The second comonad the author considers in this paper is the distributional comonad \(!_{\delta}\). Providing a symmetric monoidal storage category, which is moreover additive in an appropriate sense, with a \textit{codereliction map} modellng the differential in the context of linear logic, defines a model for intuitionistic differential linear logic. It was shown in [loc. cit.] that the category of convenient vector spaces is a model for intuitionistic differential linear logic where the comonad is the map sending a convenient vector space to the Mackey closure of the linear span of its Dirac distributions. The Kleisli category of the distribution comonad \(!_{\delta}\) is the category of convenient \(\mathcal{C}_{M}^{\infty}\)-modules and smooth morphisms. The codereliction
\[
\overline{d}_{\mathcal{E}}^{\delta}:\mathcal{E}\rightarrow!_{\delta}\mathcal{E}
\]
sends a section \(s\) to
\[
\lim_{h\rightarrow0}\,\frac{\delta_{hs}-\delta_{0}}{h}
\]
The differential of a smooth functional \(\operatorname{F}:\mathcal{E\rightarrow E}^{\prime}\) is then the bundle map
\[
dF:\mathcal{E}\multimap(\mathcal{E\Rightarrow E}^{\prime})
\]
It is shown that the two comonads \(!_{j}\) and \(!_{\delta}\) compose in the appropriate sense. It is demonstrated that the category of convenient \(\mathcal{C}_{M}^{\infty}\)-modules with the composite comonad \(!_{\delta} \circ!_{j}\) is a model for intuitionistic differential linear logic. The codereliction
\[
\overline{d}_{\mathcal{E}}^{j\delta}:\mathcal{E}\rightarrow!_{j\delta}\mathcal{E}
\]
sends a section \(s\) to
\[
\lim_{h\rightarrow0}\,\frac{\delta_{h(l(s))}-\delta_{0}}{h}
\]
In this case, a logic of smooth local functionals, knwon as Lagrangians, is obtained, where a functional is local iff the value of its variables at a point \(x\) in \(M\) depends only on its infinite jet at that point. The functional derivative of a Lagrangian \(L\) encodes the Euler-Lagrange equations as well as a total derivative, while the functional equation \(dL=0\) encodes the space of solutions to the equations of motion. Morphisms
\[
!_{\delta}!_{j}\mathcal{E}\rightarrow\mathcal{E}^{\prime}
\]
from the Kleisli category are to be interpreted as smooth differential operators. The interaction between these comonads shows how to pass between linear and non-linear objects. For linear partial differential equations with constant coefficients, some effort to understand the logic rules underlying the structure was made in [\textit{M. Kerjean}, in: Proceedings of the 2018 33rd annual ACM/IEEE symposium on logic in computer science, LICS 2018, Oxford, UK, July 9--12, 2018. New York, NY: Association for Computing Machinery (ACM). 589--598 (2018; Zbl 1452.03135)].
The paper is concluded by discussing how the above structure arises in the case when the vector bundle denotation is the trivial line bundle on an arbitrary Riemannian manifold, where the convenient \(\mathcal{C}_{M}^{\infty}\)-module is the sheaf of smooth functions on the manifold, and the variational calculus leads to the space of solutions to the scalar field equations.
Reviewer: Hirokazu Nishimura (Tsukuba)Nonassociative analogs of Lie groupoidshttps://zbmath.org/1500.220022023-01-20T17:58:23.823708Z"Grabowski, Janusz"https://zbmath.org/authors/?q=ai:grabowski.janusz"Ravanpak, Zohreh"https://zbmath.org/authors/?q=ai:ravanpak.zohrehAuthors' abstract: We introduce nonassociative geometric objects generalizing naturally Lie groupoids and called (smooth) quasiloopoids and loopoids. We prove that the tangent bundles of smooth loopoids are canonically smooth loopoids again (it is non trivial in the case of loopoids). We show also that this is not true if the cotangent bundles are concerned. After providing a few natural constructions, we show how the Lie-like functor associates with loopoids skew algebroids and almost Lie algebroids and how discrete mechanics on Lie groupoids can be reformulated in the nonassociative case.
Reviewer: J. N. Alonso Alvarez (Vigo)Symmetric ground state solutions for the Choquard Logarithmic equation with exponential growthhttps://zbmath.org/1500.340132023-01-20T17:58:23.823708Z"Yuan, Shuai"https://zbmath.org/authors/?q=ai:yuan.shuai"Chen, Sitong"https://zbmath.org/authors/?q=ai:chen.sitongSummary: We investigate the existence of ground state solutions for the fractional Choquard Logarithmic equation
\[
(- \Delta)^{1 / 2} u + V (x) u + (\ln | \cdot | \ast | u |^2) u = f (u), \quad x \in \mathbb{R},
\]
where \(V \in \mathcal{C} (\mathbb{R}, [ 0, \infty))\) and the \(f\) satisfies exponential critical growth. The present paper extends and complements the result of \textit{E. S. Böer} and \textit{O. H. Miyagaki} [``The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth'', Preprint, \url{arXiv:2011.12806}]. In particular, our paper has two typical features. Firstly, using a weaker assumption on \(f\), we establish the energy inequality to exclude the vanishing case of the required Cerami sequence. Secondly, with the property of radial symmetry we shall use some new variational and analytic technique to establish our final result which is different to the arguments explored in [loc. cit.].Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcationshttps://zbmath.org/1500.350262023-01-20T17:58:23.823708Z"Duan, Daifeng"https://zbmath.org/authors/?q=ai:duan.daifeng"Niu, Ben"https://zbmath.org/authors/?q=ai:niu.ben"Wei, Junjie"https://zbmath.org/authors/?q=ai:wei.junjieSummary: We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.On the Poincaré lemma on domainshttps://zbmath.org/1500.350962023-01-20T17:58:23.823708Z"Bousquet, Pierre"https://zbmath.org/authors/?q=ai:bousquet.pierre"Nguyen, Hoang Phuong"https://zbmath.org/authors/?q=ai:nguyen.hoang-phuongSummary: We prove the Poincaré lemma on a domain \(\Omega\) with a Dirichlet boundary condition under a natural assumption on the regularity of \(\Omega \): a closed form \(f\) in the Hölder space \(C^{r, \alpha}\) is the differential of a \(C^{r+1, \alpha}\) form, provided that the domain \(\Omega\) itself is \(C^{r+1, \alpha}\). The proof is based on a construction by approximation, together with a duality argument, in the spirit of the strategy introduced by \textit{J. Bourgain} and \textit{H. Brezis} [J. Am. Math. Soc. 16, No. 2, 393--426 (2003; Zbl 1075.35006)] to solve the divergence equation in the Sobolev space \(W_0^{1,p}(\Omega)\). We also establish the corresponding statement in the setting of higher order Sobolev spaces.On a fractional Nirenberg problem via a non-degeneracy conditionhttps://zbmath.org/1500.351502023-01-20T17:58:23.823708Z"Alghanemi, Azeb"https://zbmath.org/authors/?q=ai:alghanemi.azeb"Rigane, Afef"https://zbmath.org/authors/?q=ai:rigane.afefSummary: In this paper, we establish existence results to a fractional Nirenberg problem of order \(\sigma\in(0,\frac{3}{2})\) on the standard three dimensional
sphere. Our approach of this paper consists of studying the limits of the non-compact orbits of the gradient flow, as did Bahri for the Nirenberg problem, and performing a Morse Lemma at infinity near these ends. This involves computing their topological contributions which then yields the results.Nonnegative solution of a class of double phase problems with logarithmic nonlinearityhttps://zbmath.org/1500.351792023-01-20T17:58:23.823708Z"Aberqi, Ahmed"https://zbmath.org/authors/?q=ai:aberqi.ahmed"Benslimane, Omar"https://zbmath.org/authors/?q=ai:benslimane.omar"Elmassoudi, Mhamed"https://zbmath.org/authors/?q=ai:elmassoudi.mhamed"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: This manuscript proves the existence of a nonnegative, nontrivial solution to a class of double-phase problems involving potential functions and logarithmic nonlinearity in the setting of Sobolev space on complete manifolds. Some applications are also being investigated. The arguments are based on the Nehari manifold and some variational techniques.On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaceshttps://zbmath.org/1500.352062023-01-20T17:58:23.823708Z"Kelmer, Filipe"https://zbmath.org/authors/?q=ai:kelmer.filipe"Tenenblat, Keti"https://zbmath.org/authors/?q=ai:tenenblat.ketiSummary: We consider systems of partial differential equations of the form
\[
\begin{cases}
u_{x t} = F (u, u_x, v, v_x),\\
v_{x t} = G (u, u_x, v, v_x),
\end{cases}
\]
describing pseudospherical (\textbf{pss}) or spherical surfaces (\textbf{ss}), meaning that, their generic solutions \((u(x, t), v(x, t))\) provide metrics, with coordinates \((x, t)\), on open subsets of the plane, with constant curvature \(K = - 1\) or \(K = 1\). These systems can be described as the integrability conditions of \(\mathfrak{g}\)-valued linear problems, with \(\mathfrak{g} = \mathfrak{sl}(2, \mathbb{R})\) or \(\mathfrak{g} = \mathfrak{su}(2)\), when \(K = - 1\), \(K = 1\), respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which also contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system. We provide explicitly the first few conservation laws, from an infinite sequence, for some of the systems describing \textbf{pss}.Inverse scattering for critical semilinear wave equationshttps://zbmath.org/1500.352202023-01-20T17:58:23.823708Z"Sá Barreto, Antônio"https://zbmath.org/authors/?q=ai:sa-barreto.antonio"Uhlmann, Gunther"https://zbmath.org/authors/?q=ai:uhlmann.gunther-a"Wang, Yiran"https://zbmath.org/authors/?q=ai:wang.yiranSummary: We show that the scattering operator for defocusing energy critical semilinear wave equations \(\square u+f(u)=0\), \(f\in C^\infty(\mathbb{R})\) and \(f\sim u^5\), in three space dimensions, determines \(f\).Inheritance of generic singularities of solutions of a linear wave equation by solutions of isoentropic gas motion equationshttps://zbmath.org/1500.352452023-01-20T17:58:23.823708Z"Suleimanov, B. I."https://zbmath.org/authors/?q=ai:suleimanov.bulat-irekovich"Shavlukov, A. M."https://zbmath.org/authors/?q=ai:shavlukov.azamat-mavletovichSummary: It is shown that the catastrophe germs of smooth mappings determining the three generic (in the sense of mathematical catastrophe theory) singularities of solutions of systems of equations for a one-dimensional isoentropic gas coincide with the germs corresponding to similar singularities of solutions of a linear wave equation with constant coefficients. The conjecture is put forth that such an inheritance for generic singularities of solutions of systems of equations for a isoentropic gas must also take place in spatially multidimensional cases.Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groupshttps://zbmath.org/1500.370302023-01-20T17:58:23.823708Z"Yamaguchi, Yoshikazu"https://zbmath.org/authors/?q=ai:yamaguchi.yoshikazu.1|yamaguchi.yoshikazuAuthor's abstract: We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in \(\mathrm{PSL}_2(\mathbb{R})\). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.
Reviewer: Jialun Li (Zürich)Towards semi-classical analysis for sub-elliptic operatorshttps://zbmath.org/1500.430062023-01-20T17:58:23.823708Z"Fischer, Véronique"https://zbmath.org/authors/?q=ai:fischer.veroniqueSummary: We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on nilpotent Lie groups as well as of the notion of quantum limits in the Euclidean and nilpotent cases.
For the entire collection see [Zbl 1487.35004].The set of partial isometries as a quotient Finsler spacehttps://zbmath.org/1500.460192023-01-20T17:58:23.823708Z"Andruchow, E."https://zbmath.org/authors/?q=ai:andruchow.estebanSummary: A known general program, designed to endow the quotient space \(\mathcal{U}_{\mathcal{A}} / \mathcal{U}_{\mathcal{B}}\) of the unitary groups \(\mathcal{U}_{\mathcal{A}}, \mathcal{U}_{\mathcal{B}}\) of the \(C{}^\ast\) algebras \(\mathcal{B} \subset \mathcal{A}\) with an invariant Finsler metric, is applied to obtain a metric for the space \(\mathcal{I} (\mathcal{H})\) of partial isometries of a Hilbert space \(\mathcal{H}\). \(\mathcal{I} (\mathcal{H})\) is a quotient of the unitary group of \(\mathcal{B} (\mathcal{H}) \times \mathcal{B} (\mathcal{H})\), where \(\mathcal{B} (\mathcal{H})\) is the algebra of bounded linear operators in \(\mathcal{H}\). Under this program, the solution of a linear best approximation problem leads to the computation of minimal geodesics in the quotient space. We find solutions of this best approximation problem, and study properties of the minimal geodesics obtained.The Gromov-Hausdorff propinquity for metric spectral tripleshttps://zbmath.org/1500.460552023-01-20T17:58:23.823708Z"Latrémolière, Frédéric"https://zbmath.org/authors/?q=ai:latremoliere.fredericSummary: We define a metric on the class of metric spectral triples, which is null exactly between unitarily equivalent spectral triples. This metric dominates the propinquity, and thus implies metric convergence of the quantum compact metric spaces induced by metric spectral triples. In the process of our construction, we also introduce the covariant modular propinquity, as a key component for the definition of the spectral propinquity.Continuity of extensions of Lipschitz mapshttps://zbmath.org/1500.460592023-01-20T17:58:23.823708Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We establish the sharp rate of continuity of extensions of \(\mathbb{R}^m\)-valued 1-Lipschitz maps from a subset \(A\) of \(\mathbb{R}^n\) to a 1-Lipschitz maps on \(\mathbb{R}^n\). We consider several cases when there exists a 1-Lipschitz extension with preserved uniform distance to a given 1-Lipschitz map. We prove that, if \(m > 1\), then a given map is 1-Lipschitz and affine if and only if such a distance preserving extension exists for any 1-Lipschitz map defined on any subset of \(\mathbb{R}^n\). This shows a striking difference from the case \(m = 1\), where any 1-Lipschitz function has such a property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map \(v\) is when the difference between the two maps takes values in a fixed one-dimensional subspace of \(\mathbb{R}^m\) and the set \(A\) is geodesically convex with respect to a Riemannian pseudo-metric associated with \(v\).A bifurcation theorem related to multiple eigenvalueshttps://zbmath.org/1500.470882023-01-20T17:58:23.823708Z"You, Jia"https://zbmath.org/authors/?q=ai:you.jia"Liu, Ping"https://zbmath.org/authors/?q=ai:liu.ping"Wang, Yu Wen"https://zbmath.org/authors/?q=ai:wang.yuwen(no abstract)Transfer operators, induced probability spaces, and random walk modelshttps://zbmath.org/1500.471232023-01-20T17:58:23.823708Z"Jorgensen, P."https://zbmath.org/authors/?q=ai:jorgensen.palle-e-t"Tian, F."https://zbmath.org/authors/?q=ai:tian.fengSummary: We study a family of discrete-time random-walk models. The starting point is a fixed generalized transfer operator \(R\) subject to a set of axioms, and a given endomorphism in a compact Hausdorff space \(X\). Our setup includes a host of models from applied dynamical systems, and it leads to general path-space probability realizations of the initial transfer operator. The analytic data in our construction is a pair \((h,\lambda)\), where \(h\) is an \(R\)-harmonic function on \(X\), and \(\lambda\) is a given positive measure on \(X\) subject to a certain invariance condition defined from \(R\). With this we show that there are then discrete-time random-walk realizations in explicit path-space models; each associated to a probability measures \(\mathbb P\) on path-space, in such a way that the initial data allows for spectral characterization: The initial endomorphism in \(X\) lifts to an automorphism in path-space with the probability measure \(\mathbb P\) quasi-invariant with respect to a shift automorphism. The latter takes the form of explicit multiresolutions in \(L^2\) of \(\mathbb P\) in the sense of Lax-Phillips scattering theory.Critical points of a mean field type functional on a closed Riemann surfacehttps://zbmath.org/1500.490042023-01-20T17:58:23.823708Z"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjie"Yang, Yunyan"https://zbmath.org/authors/?q=ai:yang.yunyanSummary: Let \((\Sigma,g)\) be a closed Riemann surface and \(H^1(\Sigma)\) be the usual Sobolev space. For any real number \(\rho \), we define a generalized mean field type functional \(J_{\rho,\phi}\colon H^1(\Sigma)\rightarrow \mathbb{R}\) by
\[
J_{\rho,\phi}(u)=\frac{1}{2} \bigg(\int_{\Sigma}|\nabla_g u|^2 d v_g+\int_{\Sigma}\phi (u-\overline{u}) d v_g \bigg)-\rho\ln \int_{ \Sigma} h e^{u-\overline{u}} d v_g,
\]
where \(h\colon \Sigma\to\mathbb{R}\) is a smooth positive function, \(\phi\colon \mathbb{R}\to\mathbb{R}\) is a smooth one-variable function and \(\overline{u}=\int_\Sigma u \, d v_g/|\Sigma|\). If \(\rho\in (8k\pi,8(k+1)\pi) (k\in \mathbb{N}^\ast), \phi\) satisfies \(|\phi(t)|\leq C (|t|^p+1) (1< p< 2)\) and \(|\phi^\prime(t)|\leq C (|t|^{p-1}+1)\) for some constant \(C\), then we get critical points of \(J_{\rho,\phi}\) by adapting min-max schemes of \textit{W. Ding} et al. [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 5, 653--666 (1999; Zbl 0937.35055)], \textit{Z. Djadli} [Commun. Contemp. Math. 10, No. 2, 205--220 (2008; Zbl 1151.53035)] and \textit{A. Malchiodi} [Discrete Contin. Dyn. Syst. 21, No. 1, 277--294 (2008; Zbl 1144.35372)].Optimal control problems for an extensible beam equation with pointwise state constraintshttps://zbmath.org/1500.490092023-01-20T17:58:23.823708Z"Hwang, Jin-soo"https://zbmath.org/authors/?q=ai:hwang.jinsooSummary: This paper is devoted to deal with optimal control problems of an extensible beam equation with pointwise state constraints. By making use of Ekeland's variational principle combined with penalization and spike variation techniques, we deduce necessary optimality condition in the form of a Pontryagin's maximum principle for an observation case.Li-Yau inequalities in geometric analysishttps://zbmath.org/1500.530012023-01-20T17:58:23.823708Z"Chow, Bennett"https://zbmath.org/authors/?q=ai:chow.bennettSummary: In this short survey article, we mention some of Peter Li's tremendous influence on geometric analysis, focusing on the development of Li-Yau inequalities and related ideas in geometric analysis and geometric flows. This started with the seminal 1986 Li-Yau paper, Hamilton's development for geometric flows, and in particular his miraculous matrix Harnack estimate for the Ricci flow, Perelman's differential Harnack estimate for the conjugate heat kernel under a Ricci flow background, and his related monotonicity formulas for Ricci flow, and Bamler's recent sharp gradient estimates, new monotonicity formulas, and their applications for Ricci flow.Notes on product semisymmetric connection in a locally decomposable Riemannian spacehttps://zbmath.org/1500.530292023-01-20T17:58:23.823708Z"Maksimovic, Miroslav"https://zbmath.org/authors/?q=ai:maksimovic.miroslav-d"Stankovic, Mica"https://zbmath.org/authors/?q=ai:stankovic.mica-sSummary: The purpose of this paper is to investigate the product semisymmetric connection in a locally decomposable Riemannian space. The curvature tensors of this connection were considered. Some properties of almost product structure, some properties of torsion tensor of product semisymmetric connection and some relations between curvature tensors and almost product structure are given. Also, the paper checks a special case of such connection when its generator is a gradient vector.Para-Kähler-Einstein \(4\)-manifolds and non-integrable twistor distributionshttps://zbmath.org/1500.530322023-01-20T17:58:23.823708Z"Bor, Gil"https://zbmath.org/authors/?q=ai:bor.gil"Makhmali, Omid"https://zbmath.org/authors/?q=ai:makhmali.omid"Nurowski, Paweł"https://zbmath.org/authors/?q=ai:nurowski.pawelThis paper studies 4-manifolds with para-Kähler-Einstein metrics, which are pseudo-Riemannian metrics that come equipped with a twistor distribution on the bundle of self-dual null 2-planes. The main result is a correspondence between the anti-self dual Weyl tensor of such metrics with nonzero scalar curvature and the Cartan quartic of the associated twistor distribution. Using this correspondence, a large class of examples of twistor distributions is produced.
Reviewer: Renato G. Bettiol (New York)A formal Riemannian structure on conformal classes and uniqueness for the \(\sigma_2\)-Yamabe problemhttps://zbmath.org/1500.530422023-01-20T17:58:23.823708Z"Gursky, Matthew"https://zbmath.org/authors/?q=ai:gursky.matthew-j"Streets, Jeffrey"https://zbmath.org/authors/?q=ai:streets.jeffrey-dSummary: We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the \(\sigma_2\)-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.The Gursky-Streets equationshttps://zbmath.org/1500.530432023-01-20T17:58:23.823708Z"He, Weiyong"https://zbmath.org/authors/?q=ai:he.weiyongIn [\textit{M. Gursky} and \textit{J. Streets}, Geom. Topol. 22, No. 6, 3501--3573 (2018; Zbl 1500.53042)] a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold was introduced in the context of the \(\sigma_2\)-Yamabe problem. Recall that the \(\sigma_2\)-Yamabe problem is asking for a metric \(g_u =e^{-2u}g\) in a fixed conformal class \([g]\), such that
\[
\sigma_2 (g_u^{-1}A_u) \equiv \text{const},
\]
where \(A\) is the Schouten tensor of a given metric and \(\sigma_2\) is defined to be the \(2\)-symmetric function of the eigenvalues of \(g^{-1}A\). Establishing some regularity results for the geodesic equation, it was then proved that the solution to the \(\sigma_2\)-Yamabe problem is unique (existence of such a solution was well known). In the paper under review, the author strengthens the regularity results mentioned. For this, several new tools, such as the concavity of the Gursky-Streets operator and a convexity result for certain matrices, are established. This leads to a significant simplification of the arguments used to establish uniqueness of the solution to the \(\sigma_2\)-Yamabe problem.
Reviewer: Alexander Schmeding (Bodø)Notes about a harmonicity on the tangent bundle with vertical rescaled metrichttps://zbmath.org/1500.530502023-01-20T17:58:23.823708Z"Zagane, Abderrahım"https://zbmath.org/authors/?q=ai:zagane.abderrahim"Djaa, Nour El Houda"https://zbmath.org/authors/?q=ai:djaa.nour-elhoudaGiven a Riemannian manifold \((M,g)\), the tangent bundle \(TM\) as a manifold is equipped with a metric induced by \(g\), but with the vertical contribution of it rescaled by some positive function \(f\), giving a new metric \(G^f\). The paper is concerned with studying the harmonicity of vector fields on \(M\) with respect to the bundle metric \(G^f\). Further topics are vector fields along a mapping \(\varphi:(M,g)\to(N,h)\) with respect to the bundle metric \(H^f\), maps \((TM,G^f)\to(N,h)\), and tangent maps \((TM,G^{f_1})\to(TN,H^{f_2})\) of mappings \((M,g)\to(N,h)\). Harmonicity of all such objects is characterized by calculating the corresponding tension fields. Examples are presented, and basic theorems proven.
Reviewer: Andreas Gastel (Essen)Lichnerowicz-type estimates for the first eigenvalue of biharmonic operatorhttps://zbmath.org/1500.530522023-01-20T17:58:23.823708Z"Habibi Vosta Kolaei, Mohammad Javad"https://zbmath.org/authors/?q=ai:kolaei.mohammad-javad-habibi-vosta"Azami, Shahroud"https://zbmath.org/authors/?q=ai:azami.shahroudSummary: In this paper we are going to investigate the Lichnerowicz-type estimate theorem for the first eigenvalue of buckling and clamped plate problems on both Riemannian and Kähler manifolds. Also we will study both problems in a case of the integral curvature condition on the Kähler manifold.Closed-form geodesics and optimization for Riemannian logarithms of Stiefel and flag manifoldshttps://zbmath.org/1500.530562023-01-20T17:58:23.823708Z"Nguyen, Du"https://zbmath.org/authors/?q=ai:nguyen.du-dinhSummary: We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by \textit{K. Hüper} et al. [J. Geom. Mech. 13, No. 1, 55--72 (2021; Zbl 1477.58006)], reparameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank Stiefel manifolds. We follow the approach of minimizing the square Frobenius distance between a geodesic ending point to a given point on the manifold to compute the logarithm map and geodesic distance between two endpoints, using Fréchet derivatives to compute the gradient of this objective function. We focus on two optimization methods, \textit{gradient descent} and L-BFGS. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. The method to translate directional derivatives using Fréchet derivatives to a gradient could potentially be applied to other matrix equations.Timelike minimal surfaces in the three-dimensional Heisenberg grouphttps://zbmath.org/1500.530752023-01-20T17:58:23.823708Z"Kiyohara, Hirotaka"https://zbmath.org/authors/?q=ai:kiyohara.hirotaka"Kobayashi, Shimpei"https://zbmath.org/authors/?q=ai:kobayashi.shimpeiThe paper under review studies time-like surfaces in the three-dimensional Heisenberg group \(\operatorname{Nil}_{3}\) with left-invariant semi-Riemannian metric, \(ds^{2}_{-}\), which is defined as follows:
\[
ds^{2}_{-}=-dx^{2}_{1}+dx^{2}_{2}+\left(dx_{3}+\frac{1}{2}\left(x_{2}dx_{1}-x_{1}dx_{2}\right)\right)^{2}.
\]
The authors then use the para-complex structure for time-like surfaces to define the Abresch-Rosenberg differential and the non-linear Dirac equation with generating spinors. Then, they introduce the first result of this paper which characterizes time-like minimal surfaces in terms of the normal Gauss map, showing that non-vertical time-like minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two-sphere.
The authors also give a construction of time-like minimal surfaces in \(\operatorname{Nil}_{3}\) in terms of the para-holomorphic data, commonly, called the generalized Weierstrass-type representation. This construction is based on the loop group decomposition, mainly, the Birkhoff and Iwasawa decompositions which are in terms of the para-complex structure.
At the end of this article several examples of time-like minimal surfaces in \(\operatorname{Nil}_{3}\) in terms of para-holomorphic potentials and the generalized Weierstrass-type representation are given.
Reviewer: Hiba Bibi (Tours)Stability of Sacks-Uhlenbeck biharmonic mapshttps://zbmath.org/1500.530782023-01-20T17:58:23.823708Z"Kazemi Torbaghan, Mehdi"https://zbmath.org/authors/?q=ai:kazemi-torbaghan.seyed-mehdi"Rezaii, Morteza Mirmohammad"https://zbmath.org/authors/?q=ai:rezaii.morteza-mirmohammadSummary: In this paper, the first and second variation formulas of the Sacks-Uhlenbeck bienergy functional is obtained. As an application, instability and non-existence theorems for Sacks-Uhlenbeck biharmonic maps are given.Geometric integration by parts and Lepage equivalentshttps://zbmath.org/1500.530852023-01-20T17:58:23.823708Z"Palese, Marcella"https://zbmath.org/authors/?q=ai:palese.marcella"Rossi, Olga"https://zbmath.org/authors/?q=ai:rossi.olga"Zanello, Fabrizio"https://zbmath.org/authors/?q=ai:zanello.fabrizio.1The authors present a comparison between two approaches to the geometric formulation of calculus of variations: from one point of view the approach based on the variational complex and its representation in terms of differential forms, from the other the description in terms of variational morphisms. In both cases there are decomposition formulas which allows for a geometric description of integration by parts and the authors show how to relate the two decompositions. As a derived result, the authors introduce a recursive formula for the derivation of the Krupka-Betounes equivalents of a Lagrangian form for first-order field theories and generalize it to second-order ones. Let us outline with more details the contents of the work.
The paper is divided into 5 sections, the first one being a short introduction. The second one is a brief summary containing the definitions of the main objects which will be used in the central part of the manuscript. In Section 3 and 4 the authors present their main results, which follow from the comparison between the two approaches previously mentioned. The first and fundamental observation in Section 3 consists in the fact that any 1-contact \((n+1)\)-form can be seen as a variational morphism of codegree 0 and vice versa. Then, the authors show that the decomposition of variational morphisms in terms of a volume and a boundary terms due to Fatibene and Francaviglia, is the counterpart of the decomposition of the form using the interior Euler operator and the residual operator.
In Section 4 the previous comparison is extended to the more complicated case of 1-contact \((n-s+1)\)-form. Using the methods employed in Section 3, the authors provide a decomposition formula for these forms and introduce a local interior Euler operator and residual operator for lower degree forms. These forms can be identified with variational morphisms of codegree s and due to this identification, it is possible to compare the decomposition of variational morphisms in terms of volume and boundary terms with the one derived from the local interior Euler and residual operators: in general, they are different and further analysis is required in order to understand uniqueness and global properties of these alternative integration by parts formulas (it is important to remark that the boundary term in the decomposition of a variational morphism is not uniquely determined so that all the results presented in the paper are coherent with previous ones). Eventually, Section 5 is devoted to the presentation of a recursive formula which allows to derive the Krupka-Betounes equivalent of a Lagrangian form for first-order field theories and to extend it to second-order field theories.
The paper is well organized since the subdivision in sections and subsections is properly adapted to the contents of the work. There are introductory remarks at the beginning of each section which facilitate the reading and allow to rapidly spot the principal results. The authors have tried to make the paper self-contained adding a section containing basic notions and definitions. Nevertheless, a full understanding of the motivation and the results presented can be achieved only after reading of the main references of the paper, which are correctly mentioned whenever required. In particular, [\textit{L. Fatibene} and \textit{M. Francaviglia}, Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1138.81303); \textit{D. Krupka}, Introduction to global variational geometry. Amsterdam: Atlantis Press (2015; Zbl 1310.49001); \textit{M. Palese} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 045, 45 p. (2016; Zbl 1347.70043)] are highly recommended.
The aim of the authors is to provide a local comparison between two different approaches to geometric integration by parts, leaving global considerations to forthcoming analysis. Therefore, the proofs are mainly based on local descriptions of forms and operators. The theoretical discussion is supported by examples which are clearly presented and help the reader to connect the novelties with known results.
Reviewer: Fabio Di Cosmo (Madrid)A variational characterization of contact metric structureshttps://zbmath.org/1500.530872023-01-20T17:58:23.823708Z"Zawadzki, Tomasz"https://zbmath.org/authors/?q=ai:zawadzki.tomaszOn a manifold with a non-vanishing vector field, the squared \(L^2\)-norm of the integrability tensor of the orthogonal complement of the field is studied as a functional on the space of Riemannian metrics of fixed volume. The first variation of this action is developed for its only critical points. and the second variation of the functional at the critical points is also developed.
Reviewer: Albert Luo (Edwardsville)Evolution of eigenvalues of geometric operator under the rescaled List's extended Ricci flowhttps://zbmath.org/1500.530952023-01-20T17:58:23.823708Z"Azami, Shahroud"https://zbmath.org/authors/?q=ai:azami.shahroudSummary: Let \((M,g(t), e^{-\phi}\mathrm{d}\nu)\) be a measure space and \((g(t),\phi (t))\) evolve by the rescaled List's extended Ricci flow. In this paper, we derive the evolution equations for first eigenvalue of the geometric operators \(-\Delta_{\phi}+cS\) under the rescaled List's extended Ricci flow, where \(\Delta_{\phi}\) is the Witten Laplacian, \(\phi \in C^{\infty}(M), S=R-\alpha |\nabla \phi |^2\), and \(R\) is the scalar curvature with respect to the metric \(g(t)\). As an application, we obtain several monotonic quantities along the rescaled List's extended Ricci flow. Our results are natural extensions of some known results for Witten Laplace operator under various geometric flows.Remarks on entropy formulae for linear heat equationhttps://zbmath.org/1500.530972023-01-20T17:58:23.823708Z"Ji, Yucheng"https://zbmath.org/authors/?q=ai:ji.yuchengSummary: In this note, we prove some new entropy formula for linear heat equation on static Riemannian manifold with nonnegative Ricci curvature. The results are analogies of Cao and Hamilton's entropies for Ricci flow coupled with heat-type equations.Zeros of functionals and a parametric version of Michael selection theoremhttps://zbmath.org/1500.540052023-01-20T17:58:23.823708Z"Fomenko, T. N."https://zbmath.org/authors/?q=ai:fomenko.tatyana-nikolaevnaThe author suggests a modification of her method of search for zeros of functionals (see, e.g., [\textit{T. N. Fomenko}, Math. Notes 93, No. 1, 172--186 (2013; Zbl 1273.54053); translation from Mat. Zametki 93, No. 1, 127--143 (2013)]) and presents on that base a result on the existence, for all values of the parameter \(t\in [0;1]\), of single-valued continuous selections for a given parametric family \(F_t\) of multimaps with closed values acting from a topological space \(X\) to a complete metric space \((Y,d)\) provided such selection exists for \(F_0.\) It should be noticed that the mentioned selections are located in a given open subset of the metric space \((C(X,Y),\mu)\) of single-valued continuous maps from \(X\) to \(Y\), with the metric \(\mu(f,g):=\sup\limits_{x\in X}\,d(f(x),g(x))\).
As a consequence of this assertion, the author obtains the statement, representing a parametric version of the Michael continuous selection theorem for a family of lower semicontinuous multimaps with closed convex values.
Reviewer: Valerii V. Obukhovskij (Voronezh)Estimation of areas on some surfaces defined by the system of equationshttps://zbmath.org/1500.580012023-01-20T17:58:23.823708Z"Jabbarov, I. Sh."https://zbmath.org/authors/?q=ai:jabbarov.ilgar-shikar"Hasanova, G. K."https://zbmath.org/authors/?q=ai:hasanova.gunay-kMany problems in analysis and applied mathematics lead to the investigations of some metric questions in differentiable manifolds.
The paper under review deals with some metric questions on manifolds defined by systems of equations.
More precisely, the authors estimate areas of some surfaces defined by systems of equations.
Reviewer: Ion Mihai (Bucureşti)Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometryhttps://zbmath.org/1500.580022023-01-20T17:58:23.823708Z"Baldi, Annalisa"https://zbmath.org/authors/?q=ai:baldi.annalisa"Tesi, Maria Carla"https://zbmath.org/authors/?q=ai:tesi.maria-carla"Tripaldi, Francesca"https://zbmath.org/authors/?q=ai:tripaldi.francescaThe proof of Gaffney's inequality for differential forms goes back to [\textit{M. P. Gaffney}, Proc. Natl. Acad. Sci. USA 37, 48--50 (1951; Zbl 0042.10205)] for manifolds without boundary, and to [\textit{K. O. Friedrichs}, Commun. Pure Appl. Math. 8, 551--590 (1955; Zbl 0066.07504); \textit{C. B. Morrey jun.}, Multiple integrals in the calculus of variations. Springer, Cham (1966; Zbl 0142.38701)] for manifolds with boundary, where differential forms abide by additional conditions on the boundary. The proof of the above inequality in a compact Riemannian manifold without boundary, replacing the Sobolev space \(W^{1,2}\) by a more general Sobolev space \(W^{1,p}\), is due to [\textit{C. Scott}, Trans. Am. Math. Soc. 347, No. 6, 2075--2096 (1995; Zbl 0849.58002)] where, as a corollary, a \(L^{p}\)-Hodge decomposition result was obtained. Its counterpart, for compact Riemannian manifolds with boundary, is owing to [\textit{T. Iwaniec} et al., Ann. Mat. Pura Appl. (4) 177, 37--115 (1999; Zbl 0963.58003)]. The investigation of the \(L^{p}\)-Hodge theory for noncompact Riemannian manifolds can be seen in [\textit{E. Amar}, Math. Z. 296, No. 1--2, 877--879 (2020; Zbl 1441.58001); \textit{E. Amar}, Math. Z. 287, No. 3--4, 751--795 (2017; Zbl 1404.58005)]. An exhaustive overview of such Gaffney-type inequalities in the Riemannian setting can be found in [\textit{G. Csato} et al., J. Funct. Anal. 274, No. 2, 461--503 (2018; Zbl 1407.58002)].
The principal objective in this paper is to establish a type inequality, in \(W^{1,p}\) Sobolev spaces, for differential forms on sub-Riemannian contact manifolds, without boundary, having bounded geometry. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds [\textit{M. Rumin}, C. R. Acad. Sci., Paris, Sér. I 310, No. 6, 401--404 (1990; Zbl 0694.57010); \textit{M. Rumin}, J. Differ. Geom. 39, No. 2, 281--330 (1994; Zbl 0973.53524)], on a Sobolev-Gaffney inequality established by \textit{A. Baldi} and \textit{B. Franchi} [J. Funct. Anal. 265, No. 10, 2388--2419 (2013; Zbl 1282.35391)] in the setting of the Heisenberg groups on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.
Reviewer: Hirokazu Nishimura (Tsukuba)Classification of \(k\)-forms on \(\mathbb{R}^n\) and the existence of associated geometry on manifoldshttps://zbmath.org/1500.580032023-01-20T17:58:23.823708Z"Lê, Hông Vân"https://zbmath.org/authors/?q=ai:le-hong-van."Vanžura, Jiří"https://zbmath.org/authors/?q=ai:vanzura.jiriThis paper surveys methods and results of classification of \(k\)-forms (resp. \(k\)-vectors on \(\mathbb{R}^{n}\)), understood as description of the orbit of the standard \(\mathrm{GL}(n,\mathbb{R})\)-action on \(\Lambda ^{k}\mathbb{R}^{n}\) (resp. on \(\Lambda^{k}\mathbb{R}^{n\ast}\)). The existence of related geometry defined by differential forms on smooth manifolds is addressed. I should regrettably note in passing that the references are duplicated.
The synopsis of the paper goes as follows.
\begin{itemize}
\item[The first part of \S 2] makes several observations on the duality between \(\mathrm{GL}(n,\mathbb{R})\)-orbits of \(k\)-forms on \(\mathbb{R}^{n}\) and \(\mathrm{GL}(n,\mathbb{R})\)-orbits of \(k\)-vectors as well as the duality between \(\mathrm{GL}^{+}(n,\mathbb{R})\)-orbits of \(k\)-forms on \(\mathbb{R}^{n}\) and \(\mathrm{GL}^{+}(n,\mathbb{R})\)-orbits of \((n-k)\)-forms on \(\mathbb{R}^{n}\). The classification of \(2\)-forms on \(\mathbb{R} ^{n}\) is recalled (Theorem 2), and the Martinet's classification of \((n-2)\)-forms on \(\mathbb{R}^{n}\) [\textit{J. Martinet}, Ann. Inst. Fourier 20, No. 1, 95--178 (1970; Zbl 0189.10001)] is presented.
\item[The second part of \S 2] surveys Vinberg-Elashvili's result [\textit{È. B. Vinberg} and \textit{A. G. Èlashvili}, Sel. Math. Sov. 7, No. 1, 63--98 (1988; Zbl 0648.15021)] and some further developments by \textit{H. V. Lê} [J. Lie Theory 21, No. 2, 285--305 (2011; Zbl 1228.17025)] and \textit{H. Dietrich} et al. [J. Algebra 423, 1044--1079 (2015; Zbl 1332.17002)], which give partial information on \(\mathrm{GL}(9,\mathbb{R})\)-orbits on \(\Lambda^{3} \mathbb{R}^{9}\). Djokovic's classification of \(3\)-vectors in \(\mathbb{R}^{8}\) [\textit{D. Ž. Đoković}, Linear Multilinear Algebra 13, 3--39 (1983; Zbl 0515.15011)] is reviewed with presentation of a classification of \(5\)-forms on \(\mathbb{R}^{8}\). Djokovic's classification method combines some ideas from Vinberg-Elashvili's work and Galois cohomology method for classifying real forms of a complex orbit. Then the classification of \(\mathrm{GL}(8,\mathbb{C})\)-action on \(\Lambda^{4} \mathbb{C}^{8}\) by \textit{L. V. Antonyan} [Tr. Semin. Vektorn. Tenzorn. Anal. 20, 144--161 (1981; Zbl 0467.15018)] is reviewed, being important for classification of \(4\)-forms on \(\mathbb{R}^{8}\).
\item[\S 3] compiles known results and discusses some open problems on necessary and sufficient topological conditions for the existence of a differential \(k\)-form \(\varphi\) of given type \(\mathrm{St}_{\mathrm{GL} (n,\mathbb{R})}(\varphi(x))\) on a manifold \(M^{n}\) for \(k=2,3,4\). It is observed, in dimension \(n=8\), that the stabilizer \(\mathrm{St}_{\mathrm{GL}(n,\mathbb{R})}(\varphi)\) of a \(3\)-form \(\varphi\in\Lambda^{3}\mathbb{R}^{n\ast}\) forms a complete system of invariants of the action of \(\mathrm{GL}(n,\mathbb{R})\) on \(\mathbb{R}^{n}\).
\item[The first appendix] contains a result due to [\textit{H.-V. Le}, ``Manifolds admitting a \(\widetilde{G}_{2}\)-structure'', Preprint, \url{arXiv:0704.0503}] concerning the existence of \(3\)-form of type \(\widetilde{G}_{2}\) on a smooth \(7\)-manifold.
\item[The second appendix] (due to Mikhail Borovoi) contains the Galois cohomology method for classification of real forms of a complex orbit.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Stratified surgery and K-theory invariants of the signature operatorhttps://zbmath.org/1500.580042023-01-20T17:58:23.823708Z"Albin, Pierre"https://zbmath.org/authors/?q=ai:albin.pierre"Piazza, Paolo"https://zbmath.org/authors/?q=ai:piazza.paoloIn the paper under review, the authors generalize Higson-Roe's analytic interpretation of surgery exact sequence to the setting of stratified spaces.
More precisely, for every \(m\)-dimensional, oriented, smoothly stratified Cheeger space \(\hat {X}\) with fundamental group \(\Gamma\), they establish the following commutative diagram
\[
\begin{tikzcd}
L_{\mathrm{BQ}, d \hat{X} \times I}(\hat{X} \times I) \ar[r]\ar[d]& S_{\mathrm{BQ}}(\hat{X}) \ar[r]\ar[d]& N_{\mathrm{BQ}}(\hat{X}) \ar[r]\ar[d]&L_{\mathrm{BQ}, d \hat{X}}(\hat{X})\ar[d]\\
K_{m+1}\left(C_r^* \Gamma\right)\left[\frac{1}{2}\right] \ar[r] &K_{m+1}\left(D^*(\hat{X}_r^*)^n\right)\left[\frac{1}{2}\right] \ar[r] &K_m(\hat{X})\left[\frac{1}{2}\right] \ar[r]& K_m\left(C_r^* \Gamma\right)\left[\frac{1}{2}\right]
\end{tikzcd}
\]
Surgery theory is a subject of studying manifolds via cutting and pasting. As a sophisticated abstraction of surgery theory, surgery exact sequence [\textit{C. T. C. Wall}, Surgery on compact manifolds. 2nd ed. Providence, RI: American Mathematical Society (1999; Zbl 0935.57003)] allows people to convert concrete geometric operations into formal diagram chasing.
In a series of papers, \textit{N. Higson} and \textit{J. Roe} [\(K\)-Theory 33, No. 4, 277--299 (2004; Zbl 1083.19002); \(K\)-Theory 33, No. 4, 301--324 (2004; Zbl 1083.19003); \(K\)-Theory 33, No. 4, 325--346 (2004; Zbl 1085.19002)] established the remarkable result that there is a natural commutative diagram sending the surgery exact sequence of Wall to an ``analytic surgery sequence'' involving \(K\)-theory groups of certain \(C^*\)-algebras.
It is a meaningful task to extend the above results to singular spaces.
In the topological aspect, \textit{W. Browder} and \textit{F. Quinn} [Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 27--36 (1975; Zbl 0343.57017)] introduced a surgery exact sequence for stratified spaces (these are manifold-like singular spaces with each pure stratum being a manifold and endowed with tubular control data). In the analytic aspect, restricted to Witt or Cheeger spaces , the authors of the current paper developed the theory of signature operators [\textit{P. Albin} et al., Ann. Sci. Éc. Norm. Supér. (4) 45, No. 2, 241--310 (2012; Zbl 1260.58012); J. Noncommut. Geom. 11, No. 2, 451--506 (2017; Zbl 1375.57034); J. Reine Angew. Math. 744, 29--102 (2018; Zbl 1434.58001)]. This allows them to map the Browder-Quinn stratified surgery exact sequence to the analytic one and verify the commutativity of the diagram. A substantial technical point is the definition of ``ideal boundary condition''.
Besides the main theorem, the authors contribute a detailed account of Browder-Quinn's surgery exact sequence (which was not seen before in the literature). A geometric example on the cardinality of the stratified structure set is also included (in the spirit of [\textit{S. Chang} and \textit{S. Weinberger}, Geom. Topol. 7, 311--319 (2003; Zbl 1037.57028)]).
Reviewer: Hailiang Hu (Dalian)On the topology of moduli spaces of non-negatively curved Riemannian metricshttps://zbmath.org/1500.580052023-01-20T17:58:23.823708Z"Tuschmann, Wilderich"https://zbmath.org/authors/?q=ai:tuschmann.wilderich"Wiemeler, Michael"https://zbmath.org/authors/?q=ai:wiemeler.michaelSummary: We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.Positive solutions for slightly subcritical elliptic problems via Orlicz spaceshttps://zbmath.org/1500.580062023-01-20T17:58:23.823708Z"Cuesta, Mabel"https://zbmath.org/authors/?q=ai:cuesta.mabel"Pardo, Rosa"https://zbmath.org/authors/?q=ai:pardo.rosa-mThe paper is concerned with semilinear equations with indefinite nonlinearities in a bounded open connected st \(\Omega\subset \mathbb R^N\) for \(N>2\). More precisely, the authors investigate existence and multiplicity of positive solutions of the Dirichlet problem
\[
-\Delta u=\lambda u + a(x)f(u) \text{ in }\Omega;\quad u=0\text{ on } \partial\Omega,
\]
where \(\lambda \in \mathbb R\), the nonlinearity \(f\) satisfies a suitable growth of subcritical type and the weight \(a(x)\) does not have constant sign. Let \(\lambda_1\) be the first eigenvalue of the Laplace-Dirichlet operator on \(\Omega\). The main result states that the above Dirichlet problem admits at least a positive solution for \(\lambda \le \lambda_1\) and there exists a turning point \(\Lambda>\lambda_1\) such that the Dirichlet problem still admits at least a positive solution for \(\lambda=\Lambda\), while at least two ordered positive solutions can be found in the range \((\lambda_1,\Lambda)\). The work is inspired by \textit{S. Alama} and \textit{G. Tarantello} [Calc. Var. Partial Differ. Equ. 1, No. 4, 439--475 (1993; Zbl 0809.35022)]. The argument of the proof is based on the Crandall-Rabinowitz bifurcation theory and variational techniques, using Orlicz-Sobolev embeddings, Palais-Smale sequences and the Mountain Pass Theorem.
Reviewer: Antonio Vitolo (Fisciano)Nonexistence of the quasi-harmonic spheres and harmonic spheres into certain manifoldhttps://zbmath.org/1500.580072023-01-20T17:58:23.823708Z"Gui, Yao Ting"https://zbmath.org/authors/?q=ai:gui.yaotingThe authors prove the nonexistence of nonconstant harmonic and quasi-harmonic maps -- defined as critical points of some weighted energy functionals -- into sheres of any dimension which omit a neighbourhood of a totally geodesic submanifold of codimension 2.
Reviewer: Radu Precup (Cluj-Napoca)The relative index theorem for general first-order elliptic operatorshttps://zbmath.org/1500.580082023-01-20T17:58:23.823708Z"Bandara, Lashi"https://zbmath.org/authors/?q=ai:bandara.lashiSummary: The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as the result of Bär-Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär-Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.Higher localised \(\hat{A}\)-genera for proper actions and applicationshttps://zbmath.org/1500.580092023-01-20T17:58:23.823708Z"Guo, Hao"https://zbmath.org/authors/?q=ai:guo.hao"Mathai, Varghese"https://zbmath.org/authors/?q=ai:mathai.vargheseSummary: For a finitely generated discrete group \(\Gamma\) acting properly on a spin manifold \(M\), we formulate new topological obstructions to \(\Gamma \)-invariant metrics of positive scalar curvature on \(M\) that take into account the cohomology of the classifying space \(\underline{B}{\Gamma}\) for proper actions.
In the cocompact case, this leads to a natural generalisation of Gromov-Lawson's notion of higher \(\hat{A} \)-genera to the setting of proper actions by groups with torsion. It is conjectured that these invariants obstruct the existence of \(\Gamma \)-invariant positive scalar curvature on \(M\). For classes arising from the subring of \(H^\ast(\underline{B}{\Gamma}, \mathbb{R})\) generated by elements of degree at most 2, we are able to prove this, under suitable assumptions, using index-theoretic methods for projectively invariant Dirac operators and a twisted \(L^2\)-Lefschetz fixed-point theorem involving a weighted trace on conjugacy classes. The latter generalises a result of \textit{B.-L. Wang} and \textit{H. Wang} [J. Differ. Geom. 102, No. 2, 285--349 (2016; Zbl 1348.58014)] to the projective setting. In the special case of free actions and the trivial conjugacy class, this reduces to a theorem of \textit{V. Mathai} [Contemp. Math. 231, 203--225 (1999; Zbl 0942.46042)], which provided a partial answer to a conjecture of Gromov-Lawson on higher \(\hat{A}\)-genera. If \(M\) is non-cocompact, we obtain obstructions to \(M\) being a partitioning hypersurface inside a non-cocompact \(\Gamma \)-manifold with non-negative scalar curvature that is positive in a neighbourhood of the hypersurface. Finally, we define a quantitative version of the twisted higher index, as first introduced in [\textit{H. Guo} et al., ``Quantitative \(K\)-theory, positive scalar curvature, and band width'', Preprint, \url{arXiv:2010.01749}], and use it to prove a parameterised vanishing theorem in terms of the lower bound of the total curvature term in the square of the twisted Dirac operator.Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaceshttps://zbmath.org/1500.580102023-01-20T17:58:23.823708Z"Barraza Martínez, B."https://zbmath.org/authors/?q=ai:martinez.b-barraza|barraza-martinez.bienvenido"Denk, R."https://zbmath.org/authors/?q=ai:denk.robert"Hernández Monzón, J."https://zbmath.org/authors/?q=ai:monzon.j-hernandez|hernandez-monzon.jairo"Nendel, M."https://zbmath.org/authors/?q=ai:nendel.maxToroidal pseudodifferential calculus on the \(n\)-dimensional torus \(\mathbb{T}^n\) introduced in \textit{M. Ruzhansky} and \textit{V. Turunen}'s monograph [Pseudo-differential operators and symmetries. Background analysis and advanced topics. Basel: Birkhäuser (2010; Zbl 1193.35261)] provides us with a global quantization procedure on \(\mathbb{T}^n\). Mapping properties for such toroidal pseudodifferential operators on various kinds of function spaces such as \(L^p\), Besov and Hölder spaces have been investigated by several scholars in the scalar-valued case.
In this article, the authors study toroidal pseudodifferential operators associated with symbols with values in the space of bounded linear operators on some Banach space \(E\). Furthermore, they prove mapping properties for such pseudodifferential operators on toroidal Besov spaces with coefficients in \(E\). The symbol class used throughout the article is the class of Hörmander symbols of limited smoothness with respect to the space variable. In addition, the dyadic decomposition of the covariable space \(\mathbb{Z}^n\) is the key ingredient of the construction of \(E\)-valued Besov spaces utilized in the article. The proof of mapping properties for toroidal pseudodifferential operators between these Besov spaces is based on the convolution kernel representation of a given pseudodifferential operator using a dyadic decomposition.
Reviewer: Gihyun Lee (Ghent)Uniform resolvent estimates on manifolds of bounded curvaturehttps://zbmath.org/1500.580112023-01-20T17:58:23.823708Z"Smith, Hart F."https://zbmath.org/authors/?q=ai:smith.hart-fAuthor's abstract: We establish \(L^{q*}\rightarrow L^q\) bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of \([0,\infty)\), the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.
Reviewer: Themistocles M. Rassias (Athína)A refined version of parametric transversality theoremshttps://zbmath.org/1500.580122023-01-20T17:58:23.823708Z"Ichiki, Shunsuke"https://zbmath.org/authors/?q=ai:ichiki.shunsukeThe author upgrades the fundamental transversality results such as Thom's parametric transversality theorem and its improvement given by \textit{J. N. Mather} [Ann. Math. (2) 98, 226--245 (1973; Zbl 0242.58001)], for investigating properties of generic mappings and establishes a refined version from a new perspective of Hausdorff measures (Theorem 1.7). Some suggestive examples and comments illustrating these new results are given.
Reviewer: Dorin Andrica (Riyadh)The extra-nice dimensionshttps://zbmath.org/1500.580132023-01-20T17:58:23.823708Z"Sinha, R. Oset"https://zbmath.org/authors/?q=ai:sinha.raul-oset"Ruas, M. A. S."https://zbmath.org/authors/?q=ai:ruas.maria-aparecida-soares"Atique, R. Wik"https://zbmath.org/authors/?q=ai:atique.r-wikSummary: We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in \(C^{\infty}(N\times [0,1],P)\) is dense if and only if the pair of dimensions \((\dim N, \dim P)\) is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs \((n, p)\) for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have \({\mathscr{A}}_e\)-codimension 1 hyperplane sections. They are also related to the simplicity of \({\mathscr{A}}_e\)-codimension 2 germs. We give a sufficient condition for any \({\mathscr{A}}_e\)-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.Dirac operator spectrum on a nilmanifoldhttps://zbmath.org/1500.810442023-01-20T17:58:23.823708Z"Deandrea, Aldo"https://zbmath.org/authors/?q=ai:deandrea.aldo"Dogliotti, Fabio"https://zbmath.org/authors/?q=ai:dogliotti.fabio"Tsimpis, Dimitrios"https://zbmath.org/authors/?q=ai:tsimpis.dimitriosSummary: We obtain the spectrum of the Dirac operator on the three-dimensional Heisenberg nilmanifold \(\mathcal{M}_3\), and its complete dependence on the metric moduli. As an application, we construct the four-dimensional low-energy effective action obtained by compactification of a seven-dimensional gauge-fermion theory on \(\mathcal{M}_3\).Fractal uncertainty principle and its applications (after Bourgain, Dyatlov, Jin, Nonnenmacher, Zahl)https://zbmath.org/1500.810512023-01-20T17:58:23.823708Z"Dang, Nguyen Viet"https://zbmath.org/authors/?q=ai:dang.nguyen-vietSummary: In this talk, we will describe a new uncertainty principle that forbids to an \(L^2\) function to be localized simultaneously in space and in frequency near fractal sets satisfying certain porosity assumptions. In a second place, we will discuss spectacular applications of this principle to problems in geometric analysis on hyperbolic surfaces.
For the entire collection see [Zbl 1486.00041].\textit{pp}-wave initial datahttps://zbmath.org/1500.830092023-01-20T17:58:23.823708Z"García-Parrado, Alfonso"https://zbmath.org/authors/?q=ai:garcia-parrado-gomez-lobo.alfonsoSummary: An \textit{initial data characterization} for vacuum \textit{pp}-wave spacetimes in dimension four is constructed. This is a vacuum initial data set plus some extra conditions guaranteeing that the data development is a subset of a vacuum \textit{pp}-wave. Some of the extra conditions only depend on the same quantities used to construct the vacuum initial data, namely the \textit{first} and the \textit{second fundamental forms} while others are related to a \textit{conformal Killing initial data characterization} (CKID).Transversely trapping surfaces: dynamical versionhttps://zbmath.org/1500.830112023-01-20T17:58:23.823708Z"Yoshino, Hirotaka"https://zbmath.org/authors/?q=ai:yoshino.hirotaka"Izumi, Keisuke"https://zbmath.org/authors/?q=ai:izumi.keisuke"Shiromizu, Tetsuya"https://zbmath.org/authors/?q=ai:shiromizu.tetsuya"Tomikawa, Yoshimune"https://zbmath.org/authors/?q=ai:tomikawa.yoshimuneSummary: We propose new concepts, a dynamically transversely trapping surface (DTTS) and a marginally DTTS, as indicators for a strong gravity region. A DTTS is defined as a two-dimensional closed surface on a spacelike hypersurface such that photons emitted from arbitrary points on it in transverse directions are acceleratedly contracted in time, and a marginally DTTS is reduced to the photon sphere in spherically symmetric cases. (Marginally) DTTSs have a close analogy with (marginally) trapped surfaces in many aspects. After preparing the method of solving for a marginally DTTS in the time-symmetric initial data and the momentarily stationary axisymmetric initial data, some examples of marginally DTTSs are numerically constructed for systems of two black holes in the Brill-Lindquist initial data and in the Majumdar-Papapetrou spacetimes. Furthermore, the area of a DTTS is proved to satisfy the Penrose-like inequality \(A_0\le 4\pi (3GM)^2\), under some assumptions. Differences and connections between a DTTS and the other two concepts proposed by us previously, a loosely trapped surface [\textit{T. Shiromizu} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 3, Article ID 033E01, 6 p. (2017; Zbl 1477.83009)] and a static/stationary transversely trapping surface [\textit{H. Yoshino} et al., PTEP, Prog. Theor. Exper. Phys. 2017, No. 6, Article ID 063E01, 23 p. (2017; Zbl 1477.83070)], are also discussed. A (marginally) DTTS provides us with a theoretical tool to significantly advance our understanding of strong gravity fields. Also, since DTTSs are located outside the event horizon, they could possibly be related with future observations of strong gravity regions in dynamical evolutions.A Schwarzian on the stretched horizonhttps://zbmath.org/1500.830132023-01-20T17:58:23.823708Z"Carlip, S."https://zbmath.org/authors/?q=ai:carlip.stevenSummary: It is well known that the Euclidean black hole action has a boundary term at the horizon proportional to the area. I show that if the horizon is replaced by a stretched horizon with appropriate boundary conditions, a new boundary term appears, described by a Schwarzian action similar to the recently discovered boundary actions in ``nearly anti-de Sitter'' gravity.A mechanism of baryogenesis for causal fermion systemshttps://zbmath.org/1500.830312023-01-20T17:58:23.823708Z"Finster, Felix"https://zbmath.org/authors/?q=ai:finster.felix"Jokel, Maximilian"https://zbmath.org/authors/?q=ai:jokel.maximilian"Paganini, Claudio F."https://zbmath.org/authors/?q=ai:paganini.claudio-fSummary: It is shown that the theory of causal fermion systems gives rise to a novel mechanism of baryogenesis. This mechanism is worked out computationally in globally hyperbolic spacetimes in a way which enables the quantitative study in concrete cosmological situations.Complete classification of Friedmann-Lemaître-Robertson-Walker solutions with linear equation of state: parallelly propagated curvature singularities for general geodesicshttps://zbmath.org/1500.830382023-01-20T17:58:23.823708Z"Harada, Tomohiro"https://zbmath.org/authors/?q=ai:harada.tomohiro"Igata, Takahisa"https://zbmath.org/authors/?q=ai:igata.takahisa"Sato, Takuma"https://zbmath.org/authors/?q=ai:sato.takuma"Carr, Bernard"https://zbmath.org/authors/?q=ai:carr.bernard-jSummary: We completely classify the Friedmann-Lemaître-Robertson-Walker solutions with spatial curvature \(K = 0, \pm 1\) for perfect fluids with linear equation of state \(p = w \rho\), where \(\rho\) and \(p\) are the energy density and pressure, without assuming any energy conditions. We extend our previous work to include all geodesics and parallelly propagated (p.p.) curvature singularities, showing that no non-null geodesic emanates from or terminates at the null portion of conformal infinity and that the initial singularity for \(K = 0, -1\) and \(-5/3 < w < -1\) is a null non-scalar polynomial curvature singularity. We thus obtain the Penrose diagrams for all possible cases and identify \(w = -5/3\) as a critical value for both the future big-rip singularity and the past null conformal boundary.Self-consistent equation for torsion arising as a consequence of the Dirac sea quantum fluctuations in external classical electromagnetic and gravitational fieldshttps://zbmath.org/1500.830442023-01-20T17:58:23.823708Z"Vergeles, S. N."https://zbmath.org/authors/?q=ai:vergeles.sergei-nikitovichSummary: The quantum fluctuations of the Dirac field in external classical gravitational and electromagnetic fields are studied. A self-consistent equation for torsion is calculated, which is obtained using one-loop fermion diagrams.B-subdifferential of the projection onto the generalized spectraplexhttps://zbmath.org/1500.900712023-01-20T17:58:23.823708Z"Lin, Youyicun"https://zbmath.org/authors/?q=ai:lin.youyicun"Hu, Shenglong"https://zbmath.org/authors/?q=ai:hu.shenglongSummary: In this paper, a complete characterization of the B-subdifferential with explicit formula for the projection mapping onto the generalized spectraplex (aka generalized matrix simplex) is derived. The derivation is based on complete characterizations of the B-subdifferential as well as the Han-Sun Jacobian of the projection mapping onto the generalized simplex. The formula provides tools for further computations and nonsmooth analysis involving this projection.