東北大学 · 数学
本田章平教授の研究室は、リーマン多様体の極限空間として得られる非正則的で特異な幾何構造、特にRCD*空間やGromov–Hausdorff収束における微分幾何的性質の解明を主眼としています。特に、p-ラプラシアンの固有値の連続性や、ヘッセ形式・ラプラシアンの幾何的表現、L^p収束における微分形式の極限の定式化を展開しており、非正則空間上での微分構造の確立に貢献しています。また、熱核を用いた埋め込みによるリーマン計量の再構成や、幾何的フローの応用といった、新鋭な手法の導入も目立っています。
Figures are computed from collected data and may differ slightly.
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.
Open papers in the app to read, cite, and organize with AI.
