Weighted Poincare inequality and the Poisson equation

TL;DR

This paper establishes Green's function estimates and solves the Poisson equation on manifolds with weighted Poincare inequalities, leading to geometric and analytic insights such as Liouville properties and connectedness at infinity.

Contribution

It introduces new Green's function estimates for manifolds with weighted Poincare inequalities and applies these to solve the Poisson equation and derive geometric properties.

Findings

Green's function estimates for weighted Poincare manifolds

Existence and sharp estimates for Poisson equation solutions

Liouville property and connectedness at infinity for certain Kähler manifolds

Abstract

We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete K\"ahler manifolds. Consequently, such K\"ahler manifolds must be connected at infinity.