Stability and compactness for complete $f$-minimal surfaces

Abstract

Let $(M,\bar{g},e^{-f}d\mu )$ 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.

References in corpus (1)

• Volume growth, eigenvalue and compactness for self-shrinkers

Cited by in corpus (7)

• Eigenvalue estimate and compactness for closed $f$-minimal surfaces
• Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index
• Stability properties and gap theorem for complete $f$-minimal hypersurfaces
• Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds
• On the first stability eigenvalue of surfaces with constant weighted mean curvature
• Sobolev and isoperimetric inequalities for submanifolds in weighted ambient spaces
• Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities