Estimates for eigenvalues of Lr operator on self-shrinkers

TL;DR

This paper derives universal inequalities for the eigenvalues of a generalized differential operator on compact self-shrinkers in Euclidean space, extending previous results and analyzing cases of equality.

Contribution

It introduces new eigenvalue inequalities for the operator al L_r on self-shrinkers, generalizing prior work by Cheng and Peng.

Findings

Established universal eigenvalue inequalities for al L_r.

Extended previous eigenvalue bounds to a broader class of operators.

Analyzed conditions under which equality holds in the inequalities.

Abstract

Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$. In this paper, we study eigenvalues of the operator $\mathcal{L}_r$ on $M$, where $\mathcal{L}_r$ is defined by $$\mathcal{L}_r=e^{\frac{|x|^2}{2}}{\rm div}(e^{-\frac{|x|^2}{2}}T^r\nabla\cdot)$$ with $T^r$ denoting a positive definite (0,2)-tensor field on $M$. We obtain "universal" inequalities for eigenvalues of the operator $\mathcal{L}_r$. These inequalities generalize the result of Cheng and Peng in \cite{ChengPeng2013}. Furthermore, we also consider the case that equalities occur.