Liouville property for $f$-harmonic functions with polynomial growth

Abstract

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

arXiv commentfinal version, accepted by Kyushu J. Math