The $L^1$ Liouville Property on Weighted manifolds
  We consider a complete noncompact smooth metric measure space $(M^n,g,e^{-f} dv)$ and the associated drifting Laplacian. We find sufficient conditions on the geometry of the space so that every nonnegative $f$-subharmonic function with bounded weighted $L^1$ norm is constant. 
Nelia Charalambous and Zhiqin Lu
math.DG math.AP