Riemannian submersions with discrete spectrum

TL;DR

This paper investigates how the spectral properties of the Laplacian on a Riemannian submersion relate to the base and fibers, providing criteria for when the spectrum is discrete, especially in the case of warped products.

Contribution

It establishes new estimates and criteria linking the spectrum of the total space to the base and fiber geometries, including non-minimal fibers and warped products.

Findings

Total space spectrum relates to base and fiber spectra.

Discreteness of spectrum characterized by fiber mean curvature growth.

Warped products analyzed for spectral discreteness.

Abstract

We prove some estimates on the spectrum of the Laplacian of the total space of a Riemannian submersion in terms of the spectrum of the Laplacian of the base and the geometry of the fibers. When the fibers of the submersions are compact and minimal, we prove that the total space is discrete if and only if the base is discrete. When the fibers are not minimal, we prove a discreteness criterion for the total space in terms of the relative growth of the mean curvature of the fibers and the mean curvature of the geodesic spheres in the base. We discuss in particular the case of warped products.