Reilly-type inequalities for submanifolds in Cartan-Hadamard manifolds

TL;DR

This paper establishes new upper bounds for eigenvalues of certain operators on submanifolds in negatively curved spaces, generalizing previous results and providing Reilly-type inequalities for weighted manifolds.

Contribution

It introduces novel upper bounds for the first nonzero eigenvalues of the $p$-Laplacian and $L_T$ operator in Cartan-Hadamard manifolds, extending prior work and improving existing estimates.

Findings

New bounds for eigenvalues in negatively curved manifolds

Generalization of Niu-Xu's inequalities for $p$-Laplacian and $L_T$ operator

Reilly-type inequalities for weighted manifolds

Abstract

Let $M$ be an $m (\ge2)$-dimensional closed orientable submanifold in an $n$-dimensional complete simply-connected Riemannian manifold $N$, where the sectional curvature of $N$ is bounded above by $\delta$. When $\delta<0$, inspired by Niu-Xu (arXiv:2106.01912), we give new upper bounds for the first nonzero eigenvalues of the $p$-Laplacian and the $L_T$ operator, respectively. These generalize Niu-Xu's work for the Laplacian (arXiv:2106.01912) and improve the estimates due to Chen (Nonlinear Anal.,196, 111833, 2020) for the $p$-Laplacian and Grosjean (Hokkaido Math. J., 33(2) , 319-339, 2004) for the $L_T$ operator, respectively. We also obtain several Reilly-type inequalities for the weighted manifolds and some boundary value problems.