Stability and compactness for complete f-minimal surfaces

Original · EN

Let (M,g, e⁻ᶠdμ) be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that, in M, there is no complete two-sided Lf-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 R³ by Colding-Minicozzi.