Asian Journal of Mathematics

Volume 17 (2013)

Number 3

Volume growth, eigenvalue and compactness for self-shrinkers

Pages: 443 – 456

DOI: http://dx.doi.org/10.4310/AJM.2013.v17.n3.a3

Authors

Qi Ding (Institute of Mathematics, Fudan University, Shanghai, China)

Y. L. Xin (Institute of Mathematics, Fudan University, Shanghai, China)

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 [14]. 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.

Keywords

self-shrinkers, self similar solution, volume growth, eigenvalue estimates, compactness theorem

2010 Mathematics Subject Classification

53A07, 53A10, 53C21, 53C44

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