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

TL;DR

This paper establishes a smooth compactness theorem for $f$-minimal hypersurfaces with bounded $f$-index in certain metric measure spaces, with applications to self-shrinkers and conformal structures in low dimensions.

Contribution

It proves a new compactness theorem for $f$-minimal hypersurfaces with bounded $f$-index and volume, extending to self-shrinkers and analyzing their conformal structures.

Findings

Compactness theorem for $f$-minimal hypersurfaces with bounded $f$-index

Application to self-shrinkers in Euclidean space

Conformal structure results for $f$-minimal surfaces in 3D

Abstract

Let $(M^{n+1},g,e^{-f}d\mu)$ be a complete smooth metric measure space with $2\leq n\leq 6$ and Bakry-\'{E}mery 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 $2\leq n\leq 6$. 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.