STABILITY AND COMPACTNESS FOR COMPLETE f-MINIMAL SURFACES

Let (M, (g) over bar, e(-f) d mu) be a complete metric measure space with Bakry-Emery Ricci curvature bounded below by a positive constant. We prove that in M there is no complete two-sided L-f -stable immersed f-minimal hypersurface with finite weighted volume. Further, if M is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded f-minimal surfaces in M with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in R-3 by Colding-Minicozzi.