Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

TL;DR

This paper proves a Liouville theorem for bounded harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature, and shows the finite dimensionality of harmonic functions with polynomial growth.

Contribution

It establishes a Liouville theorem without gradient estimates and demonstrates finite dimensionality of polynomial growth harmonic functions on such solitons.

Findings

Bounded harmonic functions are constant on these Ricci solitons.

The space of polynomial growth harmonic functions has finite dimension.

Abstract

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.