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Shouhei Honda

Publications and source records attributed to Shouhei Honda.

35 records · Page 2Linked to original sources
arXiv2021

Isometric immersions of RCD spaces

We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.

math.DG↗
arXiv2020

Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces

In this short note we provide several conjectures on the regularity of measured Gromov-Hausdorff limit spaces of Riemannian manifolds with Ricci curvature bounded below, from the point of view of the synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.

math.DG↗
arXiv2019

New differential operator and non-collapsed $RCD$ spaces

We show characterizations of non-collapsed compact $RCD(K, N)$ spaces, which in particular confirm a conjecture of De Philippis-Gigli on the implication from the weakly non-collapsed condition to the non-collapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pull-back Riemannian metric by embedding in $L^2$ via the heat kernel. This seems the first application of geometric flow to the study of RCD spaces.

math.DG↗
arXiv2019

Sphere theorems for RCD and stratified spaces

We prove topological sphere theorems for RCD(n-1, n) spaces which generalize Colding's results and Petersen's result to the RCD setting. We also get an improved sphere theorem in the case of Einstein stratified spaces.

math.DG↗
arXiv2018

Bakry-Émery conditions on almost smooth metric measure spaces

In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition $BE (K, N)$. The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the $BE$ condition is strictly weaker than the $RCD$ condition even in this setting, and that the local dimension is not constant even if the space satisfies the $BE$ condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be $RCD(K, N)$ spaces.

math.DG↗
arXiv2017

Ricci curvature and Orientability

In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between $L^2$-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and the Gromov-Hausdorff convergence, which is a generalization of a recent work by Matveev-Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.

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arXiv2017

Continuity of nonlinear eigenvalues in $CD(K,\infty)$ spaces with respect to measured Gromov-Hausdorff convergence

In this note we prove in the nonlinear setting of $CD(K,\infty)$ spaces the stability of the Krasnoselskii spectrum of the Laplace operator $-Δ$ under measured Gromov-Hausdorff convergence, under an additional compactness assumption satisfied, for instance, by sequences of $CD^*(K,N)$ metric measure spaces with uniformly bounded diameter. Additionally, we show that every element $λ$ in the Krasnoselskii spectrum is indeed an eigenvalue, namely there exists a nontrivial $u$ satisfying the eigenvalue equation $- Δu = λu$.

math.MG↗
arXiv2017

Spectral convergence under bounded Ricci curvature

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are generalizations of Cheeger-Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. As a corollary we show the upper semicontinuity of the first Betti numbers with respect to the Gromov-Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. We also define the Riemannian curvature tensor, the Ricci curvature, and the scalar curvature of the limit space, and show their properties. In particular the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov-Hausdorff topology.

math.DG↗
arXiv2016

New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,d_i,m_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,d)$, we extend the results of Gigli-Mondino-Savaré by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces $H^{1,p}$, including the space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for $p>1$ and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case $N<\infty$, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of $RCD(K,\infty)$ spaces we provide stability results for Hessians and $W^{2,2}$ functions and we treat the stability of the Bakry-Émery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.

math.MG↗
arXiv2016

Fano-Ricci limit spaces and spectral convergence

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler-Ricci limit soliton and the space of all $L^2$ holomorphic vector fields with smooth potentials is a Lie algebra with respect to the Lie bracket, then the Lie algebra has the same structure as smooth Kähler-Ricci solitons. In particular if a $\Q$-Fano variety admits a Kähler-Ricci limit soliton and all holomorphic vector fields are $L^2$ with smooth potentials then the Lie algebra has the same structure as smooth Kähler-Ricci solitons. If the sequence consists of Kähler-Ricci solitons then the Ricci limit space is a weak Kähler-Ricci soliton on a $\mathbb{Q}$-Fano variety and the space of limits of $1$ eigenfunctions for the weighted $\barpartial$-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth Kähler-Ricci solitons.

math.DG↗
arXiv2015

Elliptic PDEs on compact Ricci limit spaces and applications

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. We apply these to the study of second-order differential calculus on such limit spaces.

math.DG↗
arXiv2014

Cheeger constant, $p$-Laplacian, and Gromov-Hausdorff convergence

We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds, and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also establish a new Lichnerowicz-Obata type theorem.

math.DG↗
arXiv2013

A weakly second order differential structure on rectifiable metric measure spaces

We give the definition of angles on a Gromov-Hausdorff limit space of a sequence of complete n-dimensional Riemannian manifolds with a lower Ricci curvature bound. We apply this to prove there is a weakly second order differential structure on these spaces and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

math.DG↗
arXiv2013

Ricci curvature and $L^p$-convergence

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound and to give a geometric explicit formula for the Dirichlet Laplacian on a limit space defined by Cheeger-Colding. We also prove a continuity of the first eigenvalues of the p-Laplacian with respect to the Gromov-Hausdorff topology.

math.DG↗
arXiv2010

Ricci curvature and convergence of Lipschitz functions

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and distributional Laplacian comparison theorem on limit spaces of Riemannian manifolds.

math.DG↗