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

Abstract

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.