Geometric analysis on weighted manifolds under lower $0$-weighted Ricci curvature bounds

TL;DR

This paper develops geometric analysis tools for weighted manifolds with non-negative weighted Ricci curvature, providing new eigenvalue estimates, inequalities, and analytical results.

Contribution

It introduces novel eigenvalue bounds and inequalities for weighted manifolds under zero-weighted Ricci curvature bounds, extending existing geometric analysis techniques.

Findings

First non-zero Steklov eigenvalue estimate on compact weighted manifolds with boundary.

First non-zero eigenvalue estimate on closed weighted minimal hypersurfaces.

Derived ABP estimate and Sobolev inequality of Brendle type.

Abstract

We develop geometric analysis on weighted Riemannian manifolds under lower $0$-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang-Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi-Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.