2017/12/12 by Yohei Sakurai, Sakurai, Yohei
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1712.04212
openalex publication_date 2017/12/12 · openalex created_date 2017/12/22 · openalex updated_date 2026/07/28
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.