Stability and compactness for complete $f$-minimal surfaces

Xu Cheng; Tito Mejia; Detang Zhou

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arXiv · 1210.8076

Stability and compactness for complete $f$-minimal surfaces

Xu Cheng·Tito Mejia·Detang Zhou

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Abstract

Let $(M,\bar{g}, e^{-f}dμ)$ 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.

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BibTeX RIS

Xu Cheng, Tito Mejia, Detang Zhou. 2012-10-30. Stability and compactness for complete $f$-minimal surfaces. https://arxiv.org/abs/1210.8076

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