%0 Journal Article
%T Stability and compactness for complete $f$-minimal surfaces
%A Xu Cheng
%A Tito Mejia
%A Detang Zhou
%J Mathematics
%D 2012
%I arXiv
%X Let $(M,\bar{g}, e^{-f}d\mu)$ 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 $\mathbb{R}^3$ by Colding-Minicozzi.
%U http://arxiv.org/abs/1210.8076v1