Spectral estimates and discreteness of spectra under Riemannian submersions

TL;DR

This paper investigates how the spectral properties of Riemannian submersions relate, showing that under certain conditions, the discreteness of spectra in the total and base spaces are equivalent, linking geometry and spectral theory.

Contribution

It provides new estimates for spectra under Riemannian submersions and establishes a criterion for the discreteness of spectra based on fiber geometry.

Findings

Spectrum of the base space is discrete iff the spectrum of the total space is discrete under certain conditions.

Estimates relate the spectra of total and base spaces with fiber geometry.

Discreteness of spectra is characterized by bounded mean curvature of fibers.

Abstract

For Riemannian submersions, we establish some estimates for the spectrum of the total space in terms of the spectrum of the base space and the geometry of the fibers. In particular, for Riemannian submersions of complete manifolds with closed fibers of bounded mean curvature, we show that the spectrum of the base space is discrete if and only if the spectrum of the total space is discrete.