Recent zbMATH articles in MSC 58Jhttps://zbmath.org/atom/cc/58J2024-02-15T19:53:11.284213ZWerkzeugAugmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimateshttps://zbmath.org/1526.351622024-02-15T19:53:11.284213Z"Duncan, Jonah A. J."https://zbmath.org/authors/?q=ai:duncan.jonah-a-jSummary: We obtain \textit{a priori} local pointwise second derivative estimates for solutions \(u\) to a class of augmented Hessian equations on Riemannian manifolds, in terms of the \(C^1\) norm and certain \(W^{2, p}\) norms of \(u\). We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any \(W^{2, p}\) norm can be dropped. Our results are derived using integral estimates.Corrigendum to: ``Lusternik-Schnirelman and Morse theory for the Van der Waals-Cahn-Hilliard equation with volume constraint''https://zbmath.org/1526.351822024-02-15T19:53:11.284213Z"Benci, Vieri"https://zbmath.org/authors/?q=ai:benci.vieri"Corona, Dario"https://zbmath.org/authors/?q=ai:corona.dario"Nardulli, Stefano"https://zbmath.org/authors/?q=ai:nardulli.stefano"Osorio Acevedo, Luis Eduardo"https://zbmath.org/authors/?q=ai:osorio-acevedo.luis-eduardo"Piccione, Paolo"https://zbmath.org/authors/?q=ai:piccione.paoloSummary: The statement and the proof of a technical lemma in our paper [ibid. 220, Article ID 112851, 29 p. (2022; Zbl 1490.35178)] turn out to be incorrect. Nonetheless, the main result of the paper remains valid, and in this Corrigendum we give an alternative approach which provides a correct proof of [loc. cit., Theorem 2.1].Classical solutions to the one-dimensional logarithmic diffusion equation with nonlinear Robin boundary conditionshttps://zbmath.org/1526.352072024-02-15T19:53:11.284213Z"Cortissoz, Jean C."https://zbmath.org/authors/?q=ai:cortissoz.jean-c"Reyes, César"https://zbmath.org/authors/?q=ai:reyes.cesarSummary: In this paper, we investigate the behavior of classical solutions to the one-dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely,
\[
\begin{cases}
&\partial_t u=\partial_{xx} \log u \quad\text{ in }[-l,l]\times (0, \infty)\\
& \partial_x u(\pm l, t)=\pm 2\gamma u^p(\pm l, t),
\end{cases}
\]
where \(\gamma\) is a constant. Let \(u_0 > 0\) be a smooth function defined on \([ - l, l]\), and which satisfies the compatibility condition
\[
\partial_x \log u_0(\pm l)= \pm 2\gamma u_0^{p-1}(\pm l).
\]
We show that for \(\gamma > 0\), \(p\le \frac{3}{2}\) classical solutions to the logarithmic diffusion equation above with initial data \(u_0\) are global and blow-up in infinite time, and that for \(p > 2\) there is finite time blow-up. Also, we show that in the case of \(\gamma < 0\), \(p\ge \frac{3}{2}\), solutions to the logarithmic diffusion equation with initial data \(u_0\) are global and blow-down in infinite time, but if \(p \leq 1\) there is finite time blow-down. For some of the cases mentioned above, and some particular families of examples, we provide blow-up rates and blow-down rates along sequences of times. Our approach is based on studying the Ricci flow on a cylinder endowed with a \(\mathbb{S}^1\)-symmetric metric, and some comparison arguments. Then, we bring our ideas full circle by proving a new long time existence result for the Ricci flow on a cylinder without any symmetry assumption. Finally, we show a blow-down result for the logarithmic diffusion equation on a disc.Scaling inequalities for spherical and hyperbolic eigenvalueshttps://zbmath.org/1526.352532024-02-15T19:53:11.284213Z"Langford, Jeffrey J."https://zbmath.org/authors/?q=ai:langford.jeffrey-j"Laugesen, Richard S."https://zbmath.org/authors/?q=ai:laugesen.richard-snyderSummary: Neumann and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic domains are shown to satisfy scaling inequalities or monotonicities analogous to the \(\mathrm{(length)}^{-2}\) scaling relation in Euclidean space.
For a cap of aperture \(\Theta\) on the sphere \(\mathbb{S}^2\), normalizing the \(k\)-th eigenvalue by the square of the Euclidean radius of the boundary circle yields that \(\mu_k (\Theta) \sin^2 \Theta\) is strictly decreasing, while normalizing by the stereographic radius squared gives that \(\mu_k (\Theta) 4 \tan^2 \Theta/2\) is strictly increasing. For the second Neumann eigenvalue, normalizing instead by the cap area establishes the stronger result that \(\mu_2 (\Theta) 4 \sin^2 \Theta/2\) is strictly increasing.
Monotonicities of this kind are somewhat surprising, since the Neumann eigenvalues themselves can vary non-monotonically.
Cheng and Bandle-type inequalities are deduced by assuming either fixed radius or fixed area and comparing eigenvalues of disks having different curvatures.On the second Robin eigenvalue of the Laplacianhttps://zbmath.org/1526.352542024-02-15T19:53:11.284213Z"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolong.1|li.xiaolong"Wang, Kui"https://zbmath.org/authors/?q=ai:wang.kui"Wu, Haotian"https://zbmath.org/authors/?q=ai:wu.haotianSummary: We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.Regularity of the scattering matrix for nonlinear Helmholtz eigenfunctionshttps://zbmath.org/1526.352562024-02-15T19:53:11.284213Z"Gell-Redman, Jesse"https://zbmath.org/authors/?q=ai:gell-redman.jesse"Hassell, Andrew"https://zbmath.org/authors/?q=ai:hassell.andrew"Shapiro, Jacob"https://zbmath.org/authors/?q=ai:shapiro.jacob.1|shapiro.jacob.2|shapiro.jacobSummary: We study the nonlinear Helmholtz equation \((\Delta - \lambda^2)u = \pm |u|^{p-1}u\) on \(\mathbb{R}^n, \lambda >0, p \in \mathbb{N}\) odd, and more generally \((\Delta_g + V - \lambda^2)u = N[u]\), where \(\Delta_g\) is the (positive) Laplace-Beltrami operator on an asymptotically Euclidean or conic manifold, \(V\) is a short range potential, and \(N[u]\) is a more general polynomial nonlinearity. Under the conditions \((p-1)(n-1)/2 > 2\) and \(k > (n-1)/2\), for every \(f \in H^k (\mathbb{S}^{n-1}_{\omega})\) of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form
\[
u(r, \omega) = r^{-(n-1)/2} (e^{-i\lambda r} f(\omega) + e^{+i\lambda r} b(\omega) + O(r^{-\varepsilon})), \quad \text{as } r \to \infty,
\]
for some \(b \in H^k (\mathbb{S}_{\omega}^{n-1})\) and \(\varepsilon > 0\). That is, the nonlinear scattering matrix \(f \mapsto b\) preserves Sobolev regularity, which is an improvement over the authors' previous work with \textit{J. Zhang} [SIAM J. Math. Anal. 52, No. 6, 6180--6221 (2020; Zbl 1455.35163)], that proved a similar result with a loss of four derivatives.An inverse problem for fractional connection Laplacianshttps://zbmath.org/1526.353112024-02-15T19:53:11.284213Z"Chien, Chun-Kai Kevin"https://zbmath.org/authors/?q=ai:chien.chun-kai-kevinSummary: Consider a fractional operator \(P^s, 0<s<1\), for connection Laplacian \(P:=\nabla^* \nabla +A\) on a smooth Hermitian vector bundle over a closed, connected Riemannian manifold of dimension \(n\geq 2\). We show that local knowledge of the metric, Hermitian bundle, connection, potential, and source-to-solution map associated with \(P^s\) determines these structures globally. This extends a result known for the fractional Laplace-Beltrami operator.Truncation quantization in the edge calculushttps://zbmath.org/1526.353482024-02-15T19:53:11.284213Z"Schulze, B.-Wolfgang"https://zbmath.org/authors/?q=ai:schulze.bert-wolfgang"Seiler, Jörg"https://zbmath.org/authors/?q=ai:seiler.jorgSummary: Pseudodifferential operators on the half-space associated with classical symbols of order zero without transmission property are shown to belong to the so-called edge algebra.Dimensional estimates for measures on quaternionic sphereshttps://zbmath.org/1526.420182024-02-15T19:53:11.284213Z"Ayoush, Rami"https://zbmath.org/authors/?q=ai:ayoush.rami"Wojciechowski, Michał"https://zbmath.org/authors/?q=ai:wojciechowski.michalSummary: In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansions. This estimate is an analog of a result previously obtained by the authors for the complex spheres.The resonances of the Capelli operators for small split orthosymplectic dual pairshttps://zbmath.org/1526.430082024-02-15T19:53:11.284213Z"Bramati, Roberto"https://zbmath.org/authors/?q=ai:bramati.roberto"Pasquale, Angela"https://zbmath.org/authors/?q=ai:pasquale.angela"Przebinda, Tomasz"https://zbmath.org/authors/?q=ai:przebinda.tomaszSummary: Let \((G, G')\) be a reductive dual pair in \(\mathrm{Sp}(\mathsf{W})\) with \(\operatorname{rank} G \leq \operatorname{rank} G'\) and \(G'\) semisimple. The image of the Casimir element of the universal enveloping algebra of \(G'\) under the Weil representation \(\omega\) is a Capelli operator. It is a hermitian operator acting on the smooth vectors of the representation space of \(\omega\). We compute the resonances of a natural multiple of a translation of this operator for small split orthosymplectic dual pairs. The corresponding resonance representations turn out to be GG$'$-modules in Howe's correspondence. We determine them explicitly.First BGG operators via homogeneous exampleshttps://zbmath.org/1526.530202024-02-15T19:53:11.284213Z"Gregorovič, Jan"https://zbmath.org/authors/?q=ai:gregorovic.jan"Zalabová, Lenka"https://zbmath.org/authors/?q=ai:zalabova.lenkaBernstein-Gelfand-Gelfand (BGG) complexes of operators have recently been studied intensively in connection with problems on the geometry of filtered manifolds. Early relevant contributions on the subject were [\textit{D. M. J. Calderbank} and \textit{T. Diemer}, J. Reine Angew. Math. 537, 67--103 (2001; Zbl 0985.58002); \textit{A. Čap} et al., Ann. Math. (2) 154, No. 1, 97--113 (2001; Zbl 1159.58309)]. In the present paper the authors address the case of homogeneous geometries. In particular, they provide several interesting examples of parabolic geometries leading to BGG operators. The presentation is detailed and self-contained, and the reading of the paper is suggested to non-experts to enter the field.
Reviewer: Luigi Rodino (Torino)Metrics with \(\lambda_1(-\Delta + k R) \ge 0\) and flexibility in the Riemannian Penrose inequalityhttps://zbmath.org/1526.530352024-02-15T19:53:11.284213Z"Li, Chao"https://zbmath.org/authors/?q=ai:li.chao.5"Mantoulidis, Christos"https://zbmath.org/authors/?q=ai:mantoulidis.christosGiven a closed orientable \(3\)-manifold \(M\) and the set \(\mathrm{Met}(M)\) of all smooth Riemannian metrics on \(M\), the authors consider the family
\[
\mathscr{M}_k^{\geq 0}\,=\,\{g\in\mathrm{Met}(M)\,:\,\lambda_1(-\Delta_g+kR_g)\geq 0\}\,,
\]
and the analogously defined \(\mathscr{M}_k^{> 0}\), where \(k\in\mathbb{R}\) is a constant.
Depending on the value of \(k\), the families \(\mathscr{M}_k^{\geq 0}\) and \(\mathscr{M}_k^{> 0}\) appear in various geometric contexts, such as Perelman's Ricci flow with surgery (\(k=1/4\)), the study of stable minimal hypersurfaces (\(k=1/2\)) and the Yamabe problem (\(k=1/8\)). It also makes sense to set \(k=\infty\), in which case \(\mathscr{M}_\infty^{\geq 0}\) (resp. \(\mathscr{M}_\infty^{> 0}\)) corresponds to the family of metrics with non-negative (resp. positive) scalar curvature.
Generalizing previous results in [\textit{F. C. Marques}, Ann. Math. (2) 176, No. 2, 815--863 (2012; Zbl 1319.53028)], assuming that \(M\) is topologically PSC (meaning that \(\mathscr{M}_\infty^{>0}\neq\emptyset\)) the authors prove that the moduli spaces \(\mathscr{M}_k^{\geq 0}/\mathrm{Diff}_+(M)\) and \(\mathscr{M}_k^{> 0}/\mathrm{Diff}_+(M)\) are path-connected for all \(k\in[1/4,\infty]\). Combining this with the fact that \(\mathscr{M}_\infty^{> 0}\) is contractible (proved recently in [\textit{R. H. Bamler}, \textit{B. Kleiner}, ``Ricci flow and contractibility of spaces of metrics'', Preprint, \url{arXiv:1909.08710}]), they conclude that \(\mathscr{M}_k^{\geq 0}\) and \(\mathscr{M}_k^{> 0}\) are path-connected for all \(k\in[1/4,+\infty]\). Since the case \(k=1/8\) is also of interest, the authors consider this case as well, showing contractibility of \(\mathscr{M}_{1/8}^{> 0}\) and weak contractibility of \(\mathscr{M}_{1/8}^{\geq 0}\).
The authors then apply these results to the Bartnik extension problem. It is shown that either there are no Riemannian metrics \(g\) such that \((M,g)\) admits a Bartnik extension, or \((M,g)\) admits a Bartnik extension for all \(g\in\overline{\mathscr{M}_{1/2}^{> 0}}\). A formula for the Bartnik mass in terms of the volume and topology of \(M\) is also given. Similar, stronger, conclusions are obtained for \textit{H. L. Bray}'s modification of Bartnik's mass [J. Differ. Geom. 59, No. 2, 177--267 (2001; Zbl 1039.53034)]. Finally, the authors give results in the same spirit concerning the fill-in problem proposed by Gromov in~[\textit{M. Gromov}, ``Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions'', Preprint, \url{arXiv:1811.04311}].
Reviewer: Stefano Borghini (Trento)Refinement of Hélein's conjecture on boundedness of conformal factors when \(n = 3\)https://zbmath.org/1526.530602024-02-15T19:53:11.284213Z"Plotnikov, Pavel I."https://zbmath.org/authors/?q=ai:plotnikov.pavel-i"Toland, John F."https://zbmath.org/authors/?q=ai:toland.john-francisLet \(D_{1}\) the closed unit disk centered at the origin in \(\mathbb{R}^{2}\), and \(\mathbb{S}^{2}\) the unit sphere in \(\mathbb{R}^{3}\). The purpose of the paper is to prove refinements of \textit{F. Hélein}'s conjecture in his book [Harmonic maps, conservation laws and moving frames. Transl. from the French. 2nd ed. Cambridge: Cambridge University Press (2002; Zbl 1010.58010)].
The first main result proves that if \(u\in W^{1,2}(D_{1},\mathbb{S}^{2})\) satisfies \[ \int_{D_{1}}\left\vert \partial _{1}\mathbf{n}\times \partial _{2}\mathbf{n} \right\vert dX\leq 4\pi -\delta \] for the normal vector \(\mathbf{n}\) and some \(\delta>0\), then there exist \(\Omega _{i}\in L^{2}(D_{1})\) with \[\left\Vert \Omega _{i}\right\Vert _{L^{2}(D_{1})}\leq \frac{8\pi }{ \delta }\left\Vert \nabla u\right\Vert _{L^{2}(D_{1})},\] \(i=1,2\), such that for every \(\zeta \in C_{0}^{\infty }(D_{1})\) there holds
\[\int_{D_{1}}\Phi \zeta dX=\int_{D_{1}}(\Omega _{2}\partial _{1}\zeta -\Omega _{1}\partial _{2}\zeta )dX,\]
where \(\Phi (X)=\mathbf{n}(X)\cdot (\partial _{1}\mathbf{n}(X)\times \partial _{2}\mathbf{n}(X))\).\ Moreover \(\Phi \) satisfies \(\left\Vert \Phi \right\Vert _{W^{-1,2}(D_{1})}\leq \frac{c}{\delta }\) for some constant \(c\).
For the proof, the authors first establish estimates on \(\Phi \) written as \( \Phi =\partial _{2}\omega _{1}-\partial _{1}\omega _{2}\). They introduce a parametrized family of normal vector fields and use a parameter continuation argument.
The second main result proves that if \(u\in W^{1,2}(D_{1},\mathbb{S}^{2})\) and \(\mathcal{A}\subset \mathbb{S}^{2}\) is a Borel set with positive measure \(\mu \), then there exist \(\Omega _{i}\in L^{2}(D_{1})\), \(i=1,2\), such that for every \(\zeta \in C_{0}^{\infty }(D_{1}) \)
\[\int_{D_{1}}\Phi \zeta dX=\frac{4\pi }{\mu }\int_{\mathcal{F}}\Phi \zeta dX+\int_{D_{1}}(\Omega _{2}\partial _{1}\zeta -\Omega _{1}\partial _{2}\zeta )dX,\]
and
\[\left\Vert \Omega _{i}\right\Vert _{L^{2}(D_{1})}\leq c\mu ^{-1/2}\left\Vert \nabla u\right\Vert _{L^{2}(D_{1})},\]
for some constant \(c\). Moreover \(\Phi \) satisfies \[\left\Vert \Phi -\frac{4\pi }{\mu }\chi _{ \mathcal{F}}\Phi \right\Vert _{W^{-1,2}(D_{1})}\leq c\mu ^{-1/2}\left\Vert \nabla u\right\Vert _{L^{2}(D_{1})}.\]
Here \(\mathcal{F}=\{X\in D_{1}: \mathbf{n}(X)\in \mathcal{A}\}\).
The proof relies on other results found in the paper and on the theory of regular points and a co-area formula proved by \textit{F. Bethuel} and \textit{ X.M. Zheng} [J. Funct. Anal. 80, No. 1, 60--75 (1988;Zbl 0657.46027)].
Reviewer: Alain Brillard (Riedisheim)Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metricshttps://zbmath.org/1526.530682024-02-15T19:53:11.284213Z"Lahdili, Abdellah"https://zbmath.org/authors/?q=ai:lahdili.abdellahAuthor's abstract: We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kähler metrics on a compact Kähler manifold introduced in our previous work [Proc. London Math. Soc. 119, 1065--1114 (2019; Zbl 1430.53075)]. This extends a result by \textit{R. J. Berman} and \textit{B. Berndtsson} [J. Am. Math. Soc. 30, 1165--1196 (2017; Zbl 1376.32028)] and \textit{X. X. Chen} et al. [``On deformation of extremal metrics'', Preprint, \url{arXiv:1506.01290})] in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy, thus extending our results from [Proc. London Math. Soc. 119, 1065--1114 (2019; Zbl 1430.53075)] from the polarized to the Kähler case. This implies a suitable notion of weighted \(K\)-semistability of a Kähler manifold admitting a weighted extremal Kähler metric.
Reviewer: Neda Bokan (Beograd)Curvature estimates for semi-convex solutions of Hessian equations in hyperbolic spacehttps://zbmath.org/1526.580042024-02-15T19:53:11.284213Z"Lu, Siyuan"https://zbmath.org/authors/?q=ai:lu.siyuanLet \(\sigma_k\) be the \(k\)-th elementary symmetric function, \(\nu\) be the unit outer normal, \(\kappa=(\kappa_1,\ldots,\kappa_n)\) be the principal curvatures of the hypersurface \(M\), and \(\sigma_k(\kappa)\) be the the Weingarten curvatures. The author considers the curvature estimates of the following Hessian equation.
\[
\sigma_k(\kappa(X))=f(X,\nu(X)),\text{ for all }X\in M. \tag{1}
\]
Then, the author assumes additionally that \(M\) is a closed, semi-convex, strictly star-shaped hypersurface satisfying \((1)\) in \(\mathbb H^{n+1}\), the hyperbolic space, with \(\kappa\in\Gamma_k:=\{\lambda\in\mathbb R^n:\sigma_j(\lambda)>0,1\le j\le k\}\), and considers \(f\) a positive twice continuously differentiable function on an open neighbourhood of the unit normal bundle of \(M\) in \(\mathbb H^{n+1}\times\mathbb S^{n}\). Then, he states that \(\max_{X\in M:1\le i\le n}|\kappa_i(X)|\) is bounded by a constant relying on \(n,k,\|M\|_{C^1}\), \(\inf f\), and on \(\|f\|_{C^2}\) (Theorem 1.1).
Reviewer: Mohammed El Aïdi (Bogotá)Well posedness for the Poisson problem on closed Lipschitz manifoldshttps://zbmath.org/1526.580052024-02-15T19:53:11.284213Z"Ndjinga, Michaël"https://zbmath.org/authors/?q=ai:ndjinga.michael"Nguemfouo, Marcial"https://zbmath.org/authors/?q=ai:nguemfouo.marcialSummary: We study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace-Beltrami operator is more technical than on classical differentiable manifolds (see, e.g., [\textit{F. Gesztesy} et al., J. Math. Sci., New York 172, No. 3, 279--346 (2011; Zbl 1222.58020); translation from Probl. Mat. Anal. 52, 3--58 (2010)]). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes' and Green's theorems as well as a Poincaré's inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube.Argyres-Douglas theories, modularity of minimal models and refined Chern-Simonshttps://zbmath.org/1526.580062024-02-15T19:53:11.284213Z"Kozçaz, Can"https://zbmath.org/authors/?q=ai:kozcaz.can"Shakirov, Shamil"https://zbmath.org/authors/?q=ai:shakirov.shamil"Yan, Wenbin"https://zbmath.org/authors/?q=ai:yan.wenbinSummary: The Coulomb branch indices of Argyres-Douglas theories on \(L(k, 1) \times S^1\) are recently identified with matrix elements of modular transforms of certain \(2d\) vertex operator algebras in a particular limit. A one parameter generalization of the modular transformation matrices of \((2N + 3, 2)\) minimal models are proposed to compute the full Coulomb branch index of \((A_1, A_{2N})\) Argyres-Douglas theories on the same space. Moreover, M-theory construction of these theories suggests direct connection to the refined Chern-Simons theory. The connection is made precise by showing how the modular transformation matrices of refined Chern-Simons theory are related to the proposed generalized ones for minimal models and the identification of Coulomb branch indices with the partition function of the refined Chern-Simons theory.Sub-Feller semigroups generated by pseudodifferential operators on symmetric spaces of noncompact typehttps://zbmath.org/1526.580072024-02-15T19:53:11.284213Z"Shewell Brockway, Rosemary"https://zbmath.org/authors/?q=ai:brockway.rosemary-shewellPseudo-differential operators on Lie groups represent a main stream of research at this moment, and it is natural to combine them with probability theory. In this order of ideas, basic references for the author of the present paper are the book of \textit{N. Jacob} [Pseudo differential operators and Markov processes. In 3 vol. Vol. 1: Fourier analysis and semigroups. London: Imperial College Press (2001; Zbl 0987.60003); Pseudo-differential operators and Markov processes. Vol. II: Generators and their potential theory. London: Imperial College Press (2002; Zbl 1005.60004)], devoted to applications to Markov processes in Euclidean spaces, and \textit{R. Gangolli} [Acta Math. 111, 213--246 (1964; Zbl 0154.43804)], introducing to probability measures on Lie groups. In particular, the paper is addressed to symmetric spaces of noncompact type, defined by spherical functions. In this setting pseudo-differential operators are studied, having an extension that generates a sub-Feller semigroup. Results in the Euclidean spaces are then recaptured as a particular case.
The paper originates from the Ph.D. dissertation of the author at the University of Sheffield, England, 2022, supervisor D. Applebaum.
Reviewer: Luigi Rodino (Torino)Full Laplace spectrum of distance spheres in symmetric spaces of rank onehttps://zbmath.org/1526.580082024-02-15T19:53:11.284213Z"Bettiol, Renato G."https://zbmath.org/authors/?q=ai:bettiol.renato-g"Lauret, Emilio A."https://zbmath.org/authors/?q=ai:lauret.emilio-a"Piccione, Paolo"https://zbmath.org/authors/?q=ai:piccione.paoloIt is rare that one can explicitly compute the Laplace spectrum of a closed Riemannian manifold. Even in the homogeneous setting, where techniques from representation theory can be brought to bear on the situation, explicit computation can be elusive. For instance, each sphere carries a metric of constant sectional curvature \(K >0\) and the spectrum of this metric can be computed explicitly. While, these constant curvature metrics account for the homogeneous metrics on an even-dimensional sphere, odd-dimensional spheres admit homogeneous metrics with non-constant sectional curvature, the spectra of which are unknown, in general.
Given an odd-dimensional sphere, the authors provide explicit expressions for the spectra of a two-parameter family of homogeneous metrics determined by an appropriate Hopf bundle (see Theorem~A). This is achieved by implementing a three-step representation-theoretic algorithm (developed in Sec.~2.2) for computing the spectra of the two-parameter family of homogeneous metrics in the canonical variation of a Riemannian submersion of compact normal homogeneous spaces:
\[
K/H \hookrightarrow G/H \stackrel{\pi}{\to} G/K,
\]
where \(H < K <G\) are compact Lie groups. The second step of this algorithm, which requires explicit knowledge of certain branching laws for the spherical representations determined by the pair \((G,H)\), is the main difficulty in computing the spectrum of such metrics. Fortunately, in the cases considered by the authors, the requisite branching laws are known. And, in the event \(K = HL\simeq H \times L\) for some compact Lie group \(L\), the computations can be simplified (see Cor.~2.5).
The metrics considered in Theorem~A do not account for all odd-dimensional homogeneous spheres; however, taken together with the even-dimensional spheres of constant sectional curvature, these homogeneous spheres are an exhaustive list of the distance spheres that can occur in rank-one symmetric spaces. With this in mind, the authors avail themselves of the relationship between the Laplace operator and the Jacobi operator associated to CMC hypersurfaces (see Eq.~7.2) in order to \((1)\) classify the resonant radii for distance spheres in the compact rank-one symmetric spaces \(\mathbb{C}P^{n+1}\), \(\mathbb{H}P^{n+1}\), and \(\textrm{Ca}P^{2}\), for \(n \geq 1\), and \((2)\) demonstrate that distance spheres in the non-compact rank-one symmetric spaces \(\mathbb{C} H^{n +1 }\), \(\mathbb{H}H^{n +1}\) and \(\textrm{Ca}H^{2}\), for \(n \geq 1\), are stable and locally rigid (see Theorem~B).
Reviewer: Craig Sutton (Hanover)Dirac cohomology on manifolds with boundary and spectral lower boundshttps://zbmath.org/1526.580092024-02-15T19:53:11.284213Z"Farinelli, Simone"https://zbmath.org/authors/?q=ai:farinelli.simoneSummary: Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah-Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue, which is an already known result. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant, which improves an already known result.Non-concentration of Schrödinger eigenfunctions along hypersurfaceshttps://zbmath.org/1526.580102024-02-15T19:53:11.284213Z"Wu, Xianchao"https://zbmath.org/authors/?q=ai:wu.xianchaoSummary: Let \(u_h\) be a sequence of \(L^2\)-normalized semiclassical Schrödinger eigenfunctions associated to a defect measure \(\mu\). Many recent work [\textit{C. D. Sogge}, Tôhoku Math. J. (2) 63, No. 4, 519--538 (2011; Zbl 1234.35156); \textit{C. D. Sogge} and \textit{S. Zelditch}, Princeton Math. Ser. 50, 447--461 (2014; Zbl 1307.58013); \textit{M. D. Blair} and \textit{C. D. Sogge}, J. Differ. Geom. 109, No. 2, 189--221 (2018; Zbl 1394.58016); \textit{C. Park}, Trans. Am. Math. Soc. 376, No. 8, 5809--5855 (2023; Zbl 07731971)] improved the well known \(O(h^{- \frac{1}{4}})\) restriction bounds [\textit{N. Burq} et al., Duke Math. J. 138, No. 3, 445--486 (2007; Zbl 1131.35053); \textit{R. Hu}, Forum Math. 21, No. 6, 1021--1052 (2009; Zbl 1187.35147)] under global geometric assumptions. However, in this paper we seek to improve this bound from another way. Intuitively, the more time a packet spends near a hypersurface the more concentration we would expect to see there. How to describe it quantitatively? We show that if \(\mu\) is \(\varepsilon_0\)-\textit{tangentially diffuse} with respect to the hypersurface, then one can get \(o(1)\) improvement of the well known \(O (h^{- \frac{1}{4}})\) restriction bounds.Geometric inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniqueshttps://zbmath.org/1526.580112024-02-15T19:53:11.284213Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabriceIn this survey paper, some functional inequalities are introduced for elliptic diffusion operators using curvature-dimension conditions, which include the Soblove/isoperimetric inequalities and Li-Yau type parabolic Harnack inequality. These inequalties are also extended to the sub-Riemannian setting by using generalized curvature-dimension conditions.
For the entire collection see [Zbl 1481.28001].
Reviewer: Feng-Yu Wang (Tianjin)Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifoldshttps://zbmath.org/1526.580122024-02-15T19:53:11.284213Z"Chen, Qing-Qian"https://zbmath.org/authors/?q=ai:chen.qingqian"Cheng, Li-Juan"https://zbmath.org/authors/?q=ai:cheng.lijuan"Thalmaier, Anton"https://zbmath.org/authors/?q=ai:thalmaier.antonThe authors prove that for all \( x\in D, \; f\in \mathcal{B}_b(M)\) the following estimate
\[
|Hess\; P_T f|(x)\leq \inf_{\delta>0}\left\{ \left(\frac{T}{2}\sqrt{K_1^2+\frac{K_2^2}{\delta}+\frac{2}{T}}\right)e^{T\left(K_0^-+\frac\delta2+\frac{\pi\sqrt{(n-1)K_0^-}}{2\delta_x} +\frac{\pi^2(n+3)}{4\delta^2_x}\right)}\right\}\|f\|_D
\]
holds, where \(M\) is a complete Riemannian manifold of dimension \(n\) with Levi-Civita connection \(\nabla\), \(\mathcal{B}_b(M)\) is the set of bounded measurable functions on \(M\) and \(D \subset M\) is a relatively compact open domain,\(P_T\) is the heat semigroup at the final time \(T\), \(\delta_x:=\rho(x, \partial D)\) is the Riemannain distance of \(x\) to the boundary of \(D\) and the constants;
\(K_2 := \sup\{ |d^* R+\nabla Ric)^{\#}(v,v)|(y): y\in D, v\in T_xM, |v|=1\}\),
\(K_1 := \sup\{ R(y): y\in D\}\),
\(K_0 = \inf\{ Ric(v,v): y\in D, v\in T_xM, |v|=1\}\), \(d^*\) is defined by \(d^*R(v_1)v_2:=-tr \nabla . R(., v_1)v_2\) and satisfies for all \( v_1, v_2, v_3 \in T_xM, x \in M\):
\[
\langle d^*R(v_1)v_2, v_3 \rangle = \langle (\nabla_{ v_3} Ric^\#)(v_1), v_2\rangle - \langle (\nabla_{v_2} Ric^\#)(v_3), v_1\rangle.
\]
Other estimates are established for the Hessian of the bounded positive harmonic functions \(u\) on \(D\) as follows:
\[
\begin{aligned}
|Hess\; u|(x) &\leq \inf_{0<\delta_1<\frac12, \delta_2>0 }( C_1(\delta_1, \delta_2)+ \sqrt{6} C_2(\delta_1, \delta_2)) \sqrt{\|u\|_Du(x)},\\
|Hess\; u|(x)&\leq \inf_{\delta_1, \; \delta_2>0 }( C_1(\delta_1, \delta_2)+ 2 C_2(\delta_1, \delta_2)) \|u\|_D,
\end{aligned}
\]
where \( C_1(\delta_1, \delta_2):= \sqrt{ (1+2\times\mathbf{1}_{\{K_0^-\neq 0\}})\left( (\delta_1+1) K_1^2+\frac{\delta_2}{2}K_2^2\right)}\) and \(C_2(\delta_1, \delta_2):= \frac{1}{2\delta_2} +3 K_0^-+\frac{\pi\sqrt{(n-1)K_0^-}}{2\delta_x} +\frac{\pi^2(n+3 +\delta_1^{-1})}{4\delta^2_x}\). The main proof is based on the martingale approach to the localized Bismut-Strook Hessian formula and the strong Markov property. The proof for the harmonic function is initially based on the identity \(P_t^Du=u.\)
Reviewer: Latifa Debbi (M'Sila)Reflecting Brownian motion and the Gauss-Bonnet-Chern theoremhttps://zbmath.org/1526.600462024-02-15T19:53:11.284213Z"Du, Weitao"https://zbmath.org/authors/?q=ai:du.weitao"Hsu, Elton P."https://zbmath.org/authors/?q=ai:hsu.elton-pThis paper treats the well-known Gauss-Bonnet-Chern theorem for a compact manifold with boundary. The authors adopt a probabilistic approach to use reflecting Brownian motion (RBM) in order to prove the theorem. One of the peculiar features consists in the fact that the boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.
More precisely, let \(M\) be a smooth oriented manifold with boundary, and let \(\chi (M)\) be its Euler characteristic defined by
\[
\chi (M) = \sum_{i=0}^m (-1)^i b_i,
\]
where \(b_i = \dim H^i(M)\) are the Betti numbers, and \(H^i(M)\) is the \(i\)-dimensional cohomology group. Assume that the manifold \(M\) is equipped with a Riemannian metric. A fundamental result in differential geometry is the Gauss-Bonnet-Chern (GBC) theorem, cf. [\textit{S.-S. Chern}, Ann. Math. (2) 45, 747--752 (1944; Zbl 0060.38103)] for a simple proof of the GBC theorem for a closed Riemannian manifolds. The theorem expresses the Euler characteristic as the sum of two integrals on \(M\) and on its boundary \(\partial M\)
\[
\chi(M) = \int_M e_M(x) dx + \int_{ \partial M} e_{\partial M} ( \bar{x} ) d \bar{x},
\]
where \(e_M\) (resp. \(e_{ \partial M}\) ) is a local geometric invariant determined by the curvature tensor (resp. the second fundamental form of the boundary of the manifold) respectively. For a manifold without boundary, \textit{H. P. McKean jun.} and \textit{I. M. Singer} [J. Differ. Geom. 1, 43--69 (1967; Zbl 0198.44301)] expressed the integral of the Euler form on the manifold in terms of the supertrace \(str\) of the heat kernal \(p^*\) of the Hodge-de Rham Laplacian on the bundle of differential forms:
\[
\chi(M) = \int_M str \,\,\, p^* (t,x,x ) dx.
\]
On the other hand, the probabilistic approach to analytic index theorems was initiated by \textit{J.-M. Bismut} [in: Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 491--504 (1987; Zbl 0693.58023)] for Riemannian manifolds without boundary. Roughly speaking, the above approach is based upon the fact that the heat kernel on differential forms associated with the Hodge-de Rham Laplacian of the horizontal Brownian motion on the frame bundle of the Riemannian manifold. However, Bismut's approach cannot be easily carried over to a manifold with boundary. The major difficulty lies in the fact that the multiplicative functional is discontinuous at the boundary. Another difficulty is the presence of the boundary local time, with the result that the algebraic ``fantastic cancellation (FC)'' does not occur at the path level, although FC works well in the Bismut's work. Under these circumstances, using Malliavin calculus, \textit{I. Shigekawa} et al. [Osaka J. Math. 26, No. 4, 897--930 (1989; Zbl 0704.58056)] successfully handled the singular terms on the boundary, thus giving a probabilistic proof of the Gauss-Bonnet-Chern formula. Malliavin calculus allows the heat kernel to be regarded as a generalized functional on Brownian motion which can then be expanded as an asymptotic power series in the small time parameter in a generalized sense. This probabilistic approach cannot be regarded as a direct extension of the Bismut approach because of the use of Malliavin calculus.
The purpose of the present work is to give an elementary probabilistic proof of the Gauss-Bonnet-Chern theorem without using Malliavin calculus. Their approach is heavily indebted to the previous works by \textit{E. P. Hsu} [Stochastic analysis on manifolds. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 0994.58019)] and Shigekawa et al. [loc. cit.].
Reviewer: Isamu Dôku (Saitama)Characterizing coherence with dynamical entanglementhttps://zbmath.org/1526.810092024-02-15T19:53:11.284213Z"Yang, Lian-Wu"https://zbmath.org/authors/?q=ai:yang.lianwu"Man, Zhong-Xiao"https://zbmath.org/authors/?q=ai:man.zhongxiao"Xia, Yun-Jie"https://zbmath.org/authors/?q=ai:xia.yun-jie"Cheng, Jie"https://zbmath.org/authors/?q=ai:cheng.jieSummary: Quantum coherence and entanglement are two basic aspects of nonclassicality. With the specific bipartite quantum systems which entanglement and coherence can be calculated, we provide a quantitative connection between the coherence and the dynamical entanglement that be created with the help of incoherent operations. We show that the coherence upper bounds the dynamical entanglement, and in particular conditions of the initial quantum states, the coherence of a single-party is equal to the dynamical entanglement. We extend these results to multipartite quantum systems.Propagators beyond the standard modelhttps://zbmath.org/1526.810302024-02-15T19:53:11.284213Z"Bueno Rogerio, Rodolfo José"https://zbmath.org/authors/?q=ai:bueno-rogerio.rodolfo-jose"Fabbri, Luca"https://zbmath.org/authors/?q=ai:fabbri.lucaSummary: In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.Addendum to: ``Weyl cocycles''https://zbmath.org/1526.810372024-02-15T19:53:11.284213Z"Bonora, L."https://zbmath.org/authors/?q=ai:bonora.losiano|bonora.lorianoSummary: Weyl 0- and 1-cocycles of canonical dimension 6 in six dimensions, which were computed earlier in [\textit{L. Bonora} et al., Classical Quantum Gravity 3, 635--649 (1986; Zbl 0615.58046)], are recalculated from scratch. The analysis yields five Weyl invariants (0-cocycles), instead of four, and the same four non-trivial 1-cocycles (possible trace anomalies), like in that reference (up to the correction of one typo).Squeezing and entanglement of a two-level moving atomic system for the Tavis-Cumming model via atomic coherencehttps://zbmath.org/1526.810432024-02-15T19:53:11.284213Z"Ramezani, R."https://zbmath.org/authors/?q=ai:ramezani.ramin.1|ramezani.ramin|ramezani.rohollah"Panahi, H."https://zbmath.org/authors/?q=ai:panahi.hossein|panahi.hanieh|panahi.haniyehSummary: In this paper, we consider a system of two moving two-level atoms of Tavis-Cumming model interacting with a single-mode coherent field in a lossless resonant cavity. We study the single-atom entropy squeezing, the linear entropy and the entanglement of system. We also use the Husimi distribution function and calculate the atomic Fisher information of system. Our numerical calculations indicate that the squeezing period, the squeezing time and the maximal squeezing can be controlled by appropriately choosing of the atomic motion and the field mode structure. The results show that the squeezing time and degree of entropy squeezing dependent on the initial atomic state of the atoms and the field mode. Moreover, it shows that choice of the coherent state is effective in entropy squeezing directions and by choosing the atomic coherent state as the initial state, the squeezing in x direction occurs while it was shown in [\textit{Z. Yan}, Chin. Phys. B 19, No. 7, Article ID 074207, (2010; \url{doi:10.1088/1674-1056/19/7/074207})] that only \(E\left({S}_y\right)\) can be seen. The numerical results show that the choice of the initial state as atomic coherence, the atomic motion and the field mode structure are also effective in entanglement and atomic Fisher information of the system and the atomic motion leads to a periodical time evolution of entanglement between atoms and the field and so there is a strongly dependent between the quantum entanglement and the motion factor of the atoms. We finally indicate a comparison between the atomic Fisher information and some other information entropies such as the Shannon entropy, squeezing and the linear entropy.An inner product for 4D quantum gravity and the Chern-Simons-Kodama statehttps://zbmath.org/1526.830052024-02-15T19:53:11.284213Z"Alexander, Stephon"https://zbmath.org/authors/?q=ai:alexander.stephon-h-s"Herczeg, Gabriel"https://zbmath.org/authors/?q=ai:herczeg.gabriel"Freidel, Laurent"https://zbmath.org/authors/?q=ai:freidel.laurentSummary: We demonstrate that reality conditions for the Ashtekar connection imply a non-trivial measure for the inner product of gravitational states in the polarization where the Ashtekar connection is diagonal, and we express the measure as the determinant of a certain first-order differential operator. This result opens the possibility to perform a non-perturbative analysis of the quantum gravity scalar product. In this polarization, the Chern-Simons-Kodama state, which solves the constraints of quantum gravity for a certain factor ordering, and which has de Sitter space as a semiclassical limit, is perturbatively non-normalizable with respect to the naïve inner product. Our work reopens the question of whether this state might be normalizable when the correct non-perturbative inner product and choice of integration contour are taken into account. As a first step, we perform a semi-classical treatment of the measure by evaluating it on the round three-sphere, viewed as a closed spatial slice of de Sitter. The result is a simple, albeit divergent, infinite product that might serve as a regulator for a more complete treatment of the problem. Additionally, our results suggest deep connections between the problem of computing the norm of the CSK state in quantum gravity and computing the Chern-Simons partition function for a complex group.