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

TL;DR

This paper provides sharp estimates for eigenvalues of a differential operator on self-shrinkers, including both closed and Dirichlet problems, with applications to the Ornstein-Uhlenbeck operator in Euclidean space.

Contribution

It introduces new sharp eigenvalue estimates for the operator L on self-shrinkers and extends results to the Ornstein-Uhlenbeck operator in Euclidean space.

Findings

Eigenvalue estimates for operator L on compact self-shrinkers

Sharp bounds for eigenvalues of L in bounded domains

Eigenvalue estimates for Ornstein-Uhlenbeck operator in Euclidean space

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.