Volume growth, eigenvalue and compactness for self-shrinkers

Qi Ding; Y. L. Xin

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arXiv · 1101.1411

Volume growth, eigenvalue and compactness for self-shrinkers

Qi Ding·Y. L. Xin

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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.

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Qi Ding, Y. L. Xin. 2013-10-18. Volume growth, eigenvalue and compactness for self-shrinkers. https://arxiv.org/abs/1101.1411

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