On the fundamental tone of immersions and submersions

TL;DR

This paper establishes lower bounds for the spectrum of the Laplace-Beltrami operator on certain submanifolds, extending previous estimates to a broader class of immersions and submersions with bounded mean curvature.

Contribution

It generalizes existing spectral estimates to include submanifolds in ambient spaces with Riemannian submersions over negatively curved manifolds.

Findings

Provides new lower bound estimates for the spectrum of Laplace-Beltrami operator.

Applicable to both immersions and submersions with bounded mean curvature.

Extends previous spectral bounds to more general ambient spaces.

Abstract

In this paper we obtain lower bound estimates of the spectrum of Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previous known estimates and applies for both immersions and submersions.