ESTIMATES FOR EIGENVALUES OF £ 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 [Generic mean curvature flow I; generic singularities, Ann. Math. 175 (2012) 755-833], 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 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.