Estimates for eigenvalues of Lr operator on self-shrinkers

Abstract

Let $x:M\to {\mathbb{R}}^N$ 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 $${\mathcal{L}}_r=e^{\frac{|x{|}^2}{2}}\mathrm{div}(e^{-\frac{|x{|}^2}{2}}{T}^r\nabla \cdot )$$ 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.

19 pages

References in corpus (4)

• Evolution of curves and surfaces by mean curvature
• Universal bounds on eigenvalues of the buckling problem on spherical domains
• Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds,
• Estimates for eigenvalues of the operator $L_r$