Volume growth, eigenvalue and compactness for self-shrinkers
Volume growth, eigenvalue and compactness for self-shrinkers
复制标题
DOI：
10.4310/ajm.2013.v17.n3.a3
复制
发表时间：
2011-01
期刊：
影响因子：
--
通讯作者：
Q. Ding;Y. Xin
中科院分区：
文献类型：
--
作者：
Q. Ding;Y. Xin
受Yaucite{Sy}关于单位球面上极小超曲面的第一特征值猜想的启发，证明了自缩子的最优体积增长，并估计了数学上的{L}算子的第一本征值的下界.利用特征值估计，在较弱的条件下，证明了Colding-Minicozzi得到的一类紧自缩子在{3}中的紧性定理。
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.