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 \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in \(\ir{3}\) obtained by Colding-Minicozzi under weaker conditions.