arXiv subjects

Yohei Sakurai

Publications and source records attributed to Yohei Sakurai.

25 records · Page 2Linked to original sources

arXiv2019

Upper bounds for higher-order Poincar'e constants

We introduce higher-order Poincar'e constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neumann eigenvalue problem these are related with the estimates obtained by Chung-Grigor'yan-Yau and Gozlan-Herry. We also obtain similar upper bounds for Dirichlet eigenvalues and multi-way isoperimetric constants. As an application, for manifolds with boundary of non-negative dimensional weighted Ricci curvature, we give upper bounds for inscribed radii in terms of dimension and the first Dirichlet Poincar'e constant.

math.DG↗

arXiv2019

Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type $L_p$ moment estimates of eigenfunctions.

math.DG↗

arXiv2018

Concentration of $1$-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition

In this paper, we consider a concentration of measure problem on Riemannian manifolds with boundary. We study concentration phenomena of non-negative $1$-Lipschitz functions with Dirichlet boundary condition around zero, which is called boundary concentration phenomena. We first examine relation between boundary concentration phenomena and large spectral gap phenomena of Dirichlet eigenvalues of Laplacian. We will obtain analogue of the Gromov-V. D. Milman theorem and the Funano-Shioya theorem for closed manifolds. Furthermore, to capture boundary concentration phenomena, we introduce a new invariant called the observable inscribed radius. We will formulate comparison theorems for such invariant under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. Based on such comparison theorems, we investigate various boundary concentration phenomena of sequences of manifolds with boundary.

math.MG↗

arXiv2017

Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

math.DG↗

arXiv2017

Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound

We study Riemannian manifolds with boundary under a lower $N$-weighted Ricci curvature bound for $N$ at most $1$, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity theorems for the smallest Dirichlet eigenvalues for the weighted $p$-Laplacians.

math.DG↗

arXiv2016

Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity results for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.

math.DG↗

arXiv2015

Rigidity of manifolds with boundary under a lower Ricci curvature bound

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

math.DG↗