Estimates for eigenvalues of L operator on Self-Shrinkers

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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 Rⁿ⁺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 Rⁿ⁺ᵖ. For Euclidean space Rⁿ, the differential operator L becomes the Ornstein-Uhlenbeck operator in stochastic analysis. Hence, we also give estimates for eigenvalues of the Ornstein-Uhlenbeck operator.