Stability and compactness for complete $f$-minimal surfaces

Abstract:

Let $(M,\overline {g}, e^{-f}d\mu )$ be a complete metric measure space with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove that in $M$ 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 $\mathbb {R}^3$ by Colding-Minicozzi.