Harmonic functions of polynomial growth on gradient shrinking Ricci solitons

Jia-Yong Wu, Peng Wu

Published 2019-11-07Version 1

In the previous paper (Math Ann 362:717-742, 2015), we proved that the space of $f$-harmonic functions with polynomial growth on complete non-compact gradient shrinking Ricci soliton $(M, g, f)$ is one-dimensional. In this paper, for a complete non-compact gradient shrinking Ricci soliton of bounded scalar curvature,we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We also prove analogous results for ancient ($f$-)caloric functions.