Liouville property for $f$-harmonic functions with polynomial growth
  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-\'Emery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth. 
Jia-Yong Wu
math.DG math.AP