Estimates for eigenvalues of Lr operator on self-shrinkers

Let $x:M\to {\mathbb{R}}^N$ be an $n$-dimensional compact self-shrinker in ${\mathbb{R}}^N$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. In this paper, we study eigenvalues of the operator ${\mathcal{L}}_r$ on $M$, where ${\mathcal{L}}_r$ is defined by $${\mathcal{L}}_r=e^{\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2}}\mathrm{d}\mathrm{i}\mathrm{v}(e^{-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2}}{T}^r\mathrm{\nabla }\cdot )$$ 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.