Finiteness and vanishing results on weighted Poincare inequality of complete manifolds

TL;DR

This paper investigates weighted Poincare inequalities on complete manifolds, generalizing previous results by relaxing curvature conditions and establishing finiteness of ends and vanishing of harmonic forms under specific growth conditions.

Contribution

It extends Li-Wang's results by relaxing Ricci curvature bounds and proves finiteness of ends and vanishing theorems for harmonic forms under sub-quadratic growth of the weight function.

Findings

Finiteness of ends under relaxed curvature conditions.

Vanishing of $L^2$ harmonic 1-forms with sub-quadratic weight growth.

Generalization of weighted Poincare inequality results.

Abstract

We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for $L^2$ harmonic 1-form provided that the weight function $\rho$ is of sub-quadratic growth of the distance function.