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

Abstract

Let ( M n + 1 , g , e − f d μ ) (M^{n+1},g,e^{-f}d\mu ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 2\leq n\leq 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 f -minimal hypersurfaces in M M with uniform upper bounds on f f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n + 1 \mathbb {R}^{n+1} with 2 ≤ n ≤ 6 2\leq n\leq 6 . We also prove some estimates on the f f -index of f f -minimal hypersurfaces.