Integral Ricci curvature and the mass gap of Dirichlet Laplacians on domains

TL;DR

This paper establishes a fundamental gap estimate for bounded domains in manifolds with integral Ricci curvature bounds, extending previous results to $L^p$-Ricci assumptions and utilizing geometric domain properties.

Contribution

It extends fundamental gap estimates to domains with integral Ricci curvature bounds in manifolds, introducing new techniques involving John domains.

Findings

Extended gap estimates to $L^p$-Ricci curvature bounds

Domains are shown to be John domains under these conditions

Provides eigenvalue estimates for Neumann problems

Abstract

We obtain a fundamental gap estimate for classes of bounded domains with quantitative control on the boundary in a complete manifold with integral bounds on the negative part of the Ricci curvature. This extends the result of \cite{Oden-Sung-Wang99} to $L^p$-Ricci curvature assumptions, $p>n/2$. To achieve our result, it is shown that the domains under consideration are John domains, what enables us to obtain an estimate on the first nonzero Neumann eigenvalue, which is of independent interest.