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

Abstract

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

The first version of this paper was finished in December 2017. Comments are welcome. 22 pages. A slightly revised version will appear in Hiroshima Mathematical Journal