Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature

Abstract

For Riemannian submersions with fibers of basic mean curvature, we compare the spectrum of the total space with the spectrum of a Schrödinger operator on the base manifold. Exploiting this concept, we study submersions arising from actions of Lie groups. In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups.

On the Steklov spectrum of covering spaces and total spaces

“…The proof of Theorem 1.3 relies on the methods of [22]. For convenience of the reader, we briefly discuss what will be used in the sequel.…”

“…There are various results in this direction involving compact (cf. for instance the survey [4]) or non-compact manifolds (see for example [3,6,21,22]). Our discussion is motivated by the recent paper [22], which focuses on non-compact total spaces of Riemannian principal bundles (see also [7] for the compact case).…”

“…We then say that p arises from the action of G. Similarly to the case of Riemannian coverings, we are interested in how properties of G are reflected in the spectrum of M 2 . As indicated in [22], it is natural to compare the spectrum of the Laplacian on M 2 with the spectrum of the Schrödinger operator…”

“…We denote the spectrum of the Friedrichs extension of this operator by σ L (M 1 ). Returning to our discussion on the Steklov spectrum, we begin by pointing out that if p : M 2 → M 1 is a Riemannian submersion arising from the action of a connected Lie group G, where M 1 is compact, then M 2 has bounded geometry and λ D 0 (M 2 ) > 0 (the latter one is a consequence of the recent [22,Theorem 1.1]). Therefore, we may define the Steklov spectrum of M 2 as above.…”

“…Therefore, we may define the Steklov spectrum of M 2 as above. In this setting, we establish the following analogue of [22,Theorem 1.3].…”