Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index

Abstract

Let $(M^{n+1},g,e^{-f}d\mu )$ be a complete smooth metric measure space with $2\le n\le 6$ and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded $f$-minimal hypersurfaces in $M$ with uniform upper bounds on $f$-index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in ${\mathbb{R}}^{n+1}$ with $2\le n\le 6$. We also prove some estimates on the $f$-index of $f$-minimal hypersurfaces, and give a conformal structure of $f$-minimal surface with finite $f$-index in three-dimensional smooth metric measure space.

19 pages

References in corpus (3)

• Compactness of minimal hypersurfaces with bounded index
• Stability and compactness for complete $f$-minimal surfaces
• An immersed $S^2$ self-shrinker