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

2011/12/27 by Qing-Ming Cheng, Cheng, Qing-Ming, Yejuan Peng +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #53C40 #58G25 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #math-ph #math.DG #math.MP #msc:53C40 #msc:58G25

paper · pdf · doi:10.48550/arxiv.1112.5938

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

openalex publication_date 2011/12/27 · arxiv created 2013/02/12 · arxiv updated 2013/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

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