Eigenvalue estimates for submanifolds in Hadamard manifolds and product manifolds $N\times\mathbb{R}$

TL;DR

This paper derives lower bounds for the first eigenvalues of the Laplacian and p-Laplacian on submanifolds with bounded mean curvature in Hadamard and product manifolds, advancing spectral geometry understanding.

Contribution

It provides new eigenvalue estimates for submanifolds in Hadamard and product manifolds, including bounds related to weighted fundamental tones and p-Laplacian eigenvalues.

Findings

Lower bounds for the weighted fundamental tone.

Lower bounds for the first eigenvalue of the p-Laplacian.

Eigenvalue estimates for submanifolds with bounded mean curvature.

Abstract

In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $\varphi$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p$-Laplacian.