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.
Highlights
The study of the spectrum of the Laplacian on a Riemannian manifold has attracted much attention over the last years
In order to comprehend its relations with the geometry of the underlying manifold, it is reasonable to investigate its behavior under maps between Riemannian manifolds that respect the geometry of the manifolds to some extent
In this paper, motivated by the aforementioned results, we introduce a Schrödinger operator on the base space of a Riemannian submersion with fibers of basic mean curvature and compare its spectrum with the spectrum of the total space
Summary
The study of the spectrum of the Laplacian on a Riemannian manifold has attracted much attention over the last years. Theorem 1.1 Let p : M2 → M1 be a Riemannian submersion with fibers of basic mean curvature, and consider the Schrödinger operator S as above. It is noteworthy that if the submersion has closed fibers, the operator S defined in (1) coincides with the Schrödinger operator introduced in [23], and there is a remarkable relation with the work of Bordoni [5] on Riemannian submersions with fibers of basic mean curvature Given such a submersion p : M2 → M1 with M2 closed, Bordoni considered the restrictions c and 0 of the Laplacian acting on lifted functions and on functions with zero average on any fiber, respectively, and showed in [5, Theorem 1.6] that the spectrum is written as σ (M2) = σ ( c) ∪ σ ( 0).
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