Rigidity of complete manifolds with weighted Poincar\'e inequality

TL;DR

This paper investigates the structure of complete Riemannian manifolds satisfying a weighted Poincaré inequality with Ricci curvature bounds, revealing splitting results when the weight function approaches a non-zero limit at infinity.

Contribution

It extends Li-Wang's results by analyzing manifold structure under weighted Poincaré inequalities with specific curvature conditions and weight function limits.

Findings

Manifolds exhibit splitting behavior at infinity under given conditions.

Weighted Poincaré inequality influences manifold rigidity.

Results generalize previous unweighted cases.

Abstract

We consider complete Riemannian manifolds which satisfy a weighted Poincar\`e inequality and have the Ricci curvature bounded below in terms of the weight function. When the weight function has a non-zero limit at infinity, the structure of this class of manifolds at infinity are studied and certain splitting result is obtained. Our result can be viewed as an improvement of Li-Wang's result in \cite{LW3}.