Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

Abstract

In this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on non-convex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the log-Sobolev constant on non-convex manifolds.

Gradient and Harnack inequalities on noncompact manifolds with boundary

“…We consider the convex case and pass to the nonconvex case using the conformal change of metric constructed in [Wang 2007]. Without loss of generality, we may assume that sup G := sup [0,T ]×M > 1.…”

“…To do this, they constructed cut-off functions using ρ o , the Riemannian distance function to a fixed point o ∈ M. It turns out that this argument works also when ∂ M is convex; see Section 2.1. For the nonconvex case, we will use the conformal change of metric introduced in [Wang 2007] to make a nonconvex boundary convex; see Section 2.2.…”

“…To derive explicit inequalities for the nonconvex case, we shall take a specific choice of φ as in [Wang 2007]. Let i ∂ M be the injectivity radius of ∂ M, and let ρ ∂ M be the Riemannian distance to the boundary.…”

“…By the Laplacian comparison theorem for ρ ∂ M (see [Kasue 1984, Theorem 0.3] or [Wang 2007]), (0), (0), and let…”

“…By a simple approximation argument, we may apply Theorem 1.3 and Corollary 1.4 to φ = ϕ • ρ ∂ M ; see [Wang 2007[Wang , page 1436. Obviously, (1-7) and N = ∇ρ ∂ M imply…”