Stability and compactness for complete 𝑓-minimal surfaces

Abstract

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