Poincar\'e constant on manifolds with ends

TL;DR

This paper provides optimal estimates for the Poincaré constant on manifolds with multiple ends, revealing that the second largest end influences the constant, using heat kernel estimates and extending previous results.

Contribution

It introduces new heat kernel estimates for parabolic manifolds with ends and shows the Poincaré constant depends on the second largest end, which is a novel insight.

Findings

Poincaré constant is determined by the second largest end.

Extended heat kernel estimates to a broader class of manifolds.

Provided optimal bounds for Poincaré constants on manifolds with ends.

Abstract

We obtain optimal estimates of the Poincar\'e constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincar\'e constant is determined by the second largest end. The proof is based on the argument by Kusuoka-Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.