Volume growth eigenvalue and compactness for self-shrinkers

Abstract

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\mathbb{R}^3$ obtained by Colding-Minicozzi under weaker conditions.