Stability and compactness for complete f-minimal surfaces

2012/10/30 by Cheng, Xu, Mejia, Tito, Zhou, Detang · 2 citations
#49Q05 #53A10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1210.8076

Abstract

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 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 ℝ³ by Colding-Minicozzi.

Cited by

• Stability of weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound
• Self-shrinkers with any number of ends in ℝ³ by stacking ℝ²

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• Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor