Volume growth, eigenvalue and compactness for self-shrinkers

TL;DR

This paper establishes optimal volume growth and eigenvalue bounds for self-shrinkers, leading to a compactness theorem for certain classes of these geometric objects, advancing understanding in geometric analysis.

Contribution

It provides the first optimal volume growth results and eigenvalue estimates for self-shrinkers, and proves a new compactness theorem under weaker conditions.

Findings

Optimal volume growth for self-shrinkers

Lower bound estimate for the first eigenvalue of the operator

Compactness theorem for a class of self-shrinkers in
3

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.