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

Abstract

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.

21 pages, a final version, accepted for publication in Communications in Contemporary Mathematics

References in corpus (5)

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Cited by in corpus (2)

• Complete self-shrinkers of the mean curvature flow
• On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space II