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

In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator $L$, which is introduced by Colding and Minicozzi in [4], on an $n$-dimensional compact self-shrinker in ${R}^{n+p}$. Estimates for eigenvalues of the differential operator $L$ are obtained. Our estimates for eigenvalues of the differential operator $L$ are sharp. Furthermore, we also study the Dirichlet eigenvalue problem of the differential operator $L$ on a bounded domain with a piecewise smooth boundary in an $n$-dimensional complete self-shrinker in ${R}^{n+p}$. For Euclidean space ${R}^{n}$, the differential operator $L$ becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator.