Lp Liouville type theorems for harmonic functions on gradient Ricci solitons

Abstract

In this paper we consider $L^p$ Liouville type theorems for harmonic functions on gradient Ricci solitons. In particular, assume that $(M,g)$ is a gradient shrinking or steady Kähler-Ricci soliton, then we prove that any pluriharmonic function u on M with $\nabla u\in L^p\left(M\right)$ for some $1<p\le 2$ is a constant function.