Stability and compactness for complete f -minimal surfaces

By Xu Cheng, Tito Mejia, Detang Zhou. 2012

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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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb74e0b38e7e73d601464a93ed660329b1a6881d