Estimates for eigenvalues of Lr operator on self-shrinkers

Let $x: M\rightarrow \mathbb{R}^{N}$ be an $n$-dimensional compact self-shrinker in $\mathbb{R}^N$ with smooth boundary $\partialΩ$. 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.

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