The $L^1$ Liouville Property on Weighted manifolds

TL;DR

This paper establishes geometric conditions on weighted manifolds ensuring that nonnegative subharmonic functions with bounded weighted $L^1$ norm are constant, extending Liouville-type results.

Contribution

It provides new sufficient conditions on the geometry of weighted manifolds for the Liouville property of the drifting Laplacian.

Findings

Nonnegative $f$-subharmonic functions with bounded weighted $L^1$ norm are constant under certain geometric conditions.

The results extend classical Liouville theorems to weighted manifolds with the drifting Laplacian.

The paper identifies specific geometric criteria that guarantee the Liouville property for the weighted setting.

Abstract

We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with bounded weighted $L^1$ norm is constant.