Estimates for eigenvalues of L operator on Self-Shrinkers

Details Estimates for eigenvalues of L operator on Self-Shrinkers Estimates for eigenvalues of L operator on Self-Shrinkers

2011-12-27

arxiv.org/abs/1112.5938v2

Mathematics

Differential Geometry

16 pages, revised version

Scientific paper

In this paper we study eigenvalues of the differential operator $\mathfrak L$, which is introduced by Colding and Minicozzi in \cite{CM1}, on an $n$-dimensional compact self-shrinker without boundary in $\mathbf {R}^{n+p}$ and on a bounded domain in an $n$-dimensional complete self-shrinker in $\mathbf {R}^{n+p}$. Our estimates for eigenvalues of the differential operator $\mathfrak L$ are sharp. For Euclidean space $\mathbf {R}^{n}$, the differential operator $\mathfrak L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator.

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