For a Riemannian covering p : M 2 → M 1 , we compare the spectrum of an essentially self-adjoint differential operator D 1 on a bundle E 1 → M 1 with the spectrum of its lift D 2 on p * E 1 → M 2 . We prove that if the covering is infinite sheeted and amenable, then the spectrum of D 1 is contained in the essential spectrum of any self-adjoint extension of D 2 . We show that if the deck transformations group of the covering is infinite and D 2 is essentially self-adjoint (or symmetric and bounded from below), then D 2 (or the Friedrichs extension of D 2 ) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M 1 is closed, then p is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.
For a Riemannian covering M 1 → M 0 of connected Riemannian manifolds with respective fundamental groups 1 ⊆ 0 , we show that the bottoms of the spectra of M 0 and M 1 coincide if the right action of 0 on 1 \ 0 is amenable.
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.
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.
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