Sharp Reilly-type inequalities for submanifolds in space forms

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by $L_{T}f=-div(T\nabla f)$, where $T$ is a general symmetric, positive definite and divergence-free $(1,1)$-tensor on $M$. The upper bound is given in terms of an integration involving $tr T$ and $|H_T|^2$, where $tr T$ is the trace of the tensor $T$ and $H_T=\sum_{i=1}^nA(Te_i,e_i)$ is a normal vector field associated with $T$ and the second fundamental form $A$ of $M$. Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous `Reilly inequality'. The operator $L_{T}$ can be regarded as a natural generalization of the well-known operator $L_r$ which is the linearized operator of the first variation of the $(r+1)$-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of Grosjean ([16]) and Li-Wang ([19]) for hypersurfaces to higher codimension case.