On $L^p$-Liouville property for smooth metric measure spaces

TL;DR

This paper investigates the $L_f^p$-Liouville property for nonnegative $f$-subharmonic functions on complete noncompact smooth metric measure spaces, establishing sharp conditions under various curvature bounds and growth constraints.

Contribution

It provides new sharp $L_f^p$-Liouville theorems for spaces with bounded below $ ext{Ric}_f^m$ and nonnegative $ ext{Ric}_f$, extending previous results to the case $0<p<1$.

Findings

Proved a sharp $L_f^p$-Liouville theorem for $0<m<inite$.

Established an $L_f^p$-Liouville theorem under $ ext{Ric}_f extgreater=0$ with controlled $f$ growth.

Extended Liouville properties to the case $0<p<1$ for nonnegative $f$-subharmonic functions.

Abstract

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with $\mathrm{Ric}_f^m$ bounded below for $0<m\leq\infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $L_f^p$-Liouville theorem when $\mathrm{Ric}_f\geq 0$ and $|f(x)|\leq \delta(n) \ln r(x)$.