Tohoku University · Mathematics
Professor Shouhei Honda's research lab specializes in geometric analysis and metric measure geometry, focusing on the structure and convergence of Riemannian and RCD spaces under Gromov–Hausdorff topology. The lab investigates the behavior of differential structures, Laplacians, and p-Laplacians on limit spaces arising from sequences of manifolds with Ricci curvature bounds, particularly in noncollapsed settings. Key directions include L^p-convergence of tensor fields, spectral stability, and the application of heat kernel embeddings and geometric flows to understand curvature and measure-theoretic properties of singular spaces.
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Abstract 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 the cont
We show characterizations of noncollapsed compact [math] spaces, which in particular confirm a conjecture of De Philippis and Gigli on the implication from the weakly noncollapsed condition to the noncollapsed one in the compact case. The key idea is to give the explicit formula of the Laplacian associated to the pullback Riemannian metric by embedding in [math] via the heat kernel. This seems to be the first application of geometric flow to the study of [math] spaces.
We give the definition of a convergence of the differentials of Lipschitz functions with respect to the measured Gromov-Hausdorff topology and several properties of the convergence.
In this paper we study the family of embeddings Φ<sub>t</sub> of a compact RCD<sup>⁎</sup>(K,N) space (X,d,m) into L<sup>2</sup>(X,m) via eigenmaps. Extending part of the classical results [10,11] known for closed Riemannian manifolds, we prove convergence as t↓0 of the rescaled pull-back metrics Φ<sub>t</sub><sup>⁎</sup>g<sub>L<sup>2</sup></sub> in L<sup>2</sup>(X,m) induced by Φ<sub>t</sub>. Moreover we discuss the behavior of Φ<sub>t</sub><sup>⁎</sup>g<sub>L<sup>2</sup></sub> with respect to
A weakly second-order differential structure on rectifiable metric measure spaces
In this paper, we study limit spaces of a sequence of n-dimensional complete Riemannian manifolds whose Ricci curvatures have definite lower bound. We will give several measure theoretical properties of such limit spaces.
We study harmonic functions with polynomial growth on asymptotic cones of a nonnegatively Ricci curved manifold with Euclidean volume growth. Especially, we will give the classification of such harmonic functions.
Abstract 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 condi
In this paper, we give a generalization of Cheeger-Colding's suspension theorem for manifolds with almost maximal diameters. We also discuss a relationship between the eigenvalues of the Laplacian and the structure of tangent cones of non-collapsing limit spaces.
International audience
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.
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