1112.5938 (Qing-Ming Cheng et al.)

Estimates for eigenvalues of $\mathfrak L$ operator on Self-Shrinkers    [PDF]

Qing-Ming Cheng, Yejuan Peng

In this paper we study eigenvalues of the closed eigenvalue problem of the differential operator $\mathfrak L$, which is introduced by Colding and Minicozzi in \cite{CM1}, on an $n$-dimensional compact self-shrinker in $\mathbf {R}^{n+p}$. Estimates for eigenvalues of the differential operator $\mathfrak L$ are obtained. Our estimates for eigenvalues of the differential operator $\mathfrak L$ are sharp. As an application of our estimates for eigenvalues, we give an optimal upper bound for the first eigenvalue and a characterization of compact self-shrinkers in $\mathbf {R}^{n+p}$ with arbitrary co-dimension is given by the first eigenvalue. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator $\mathfrak L$ on a bounded domain with a piecewise smooth boundary in an $n$-dimensional complete self-shrinker in $\mathbf {R}^{n+p}$. 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.

View original: http://arxiv.org/abs/1112.5938