Spectral estimates and discreteness of spectra under Riemannian submersions

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.

Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature

“…It is noteworthy that if the submersion has closed fibers, then 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 : M 2 → M 1 with M 2 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 (M 2 ) = σ ( c ) ∪ σ ( 0 ).…”

“…Recently, in [23], extending the result of [10], we established a lower bound for the bottom of the spectrum λ 0 (M 2 ) of M 2 , if the mean curvature of the fibers is bounded in a certain way. More precisely, according to [23,Theorem 1.1], if the (unnormalized) mean curvature of the fibers is bounded by H ≤ C ≤ 2 √ λ 0 (M 1 ), then the bottom of the spectrum of M 2 satisfies…”

“…In this context, we introduced a Schrödinger operator on M 1 , with potential determined by the volume of the fibers, and compared its spectrum with the spectrum of M 2 . It should be noticed that if the submersion has fibers of infinite volume, then we are not able to define that operator, at least in the way we did in [23].…”

“…Recall that, in general, λ 0 (F x ) is only upper semi-continuous with respect to x ∈ M 1 (cf. [23,Lemma 2.9]).…”

“…In the second part of [23], following [4], we studied Riemannian submersions with closed fibers. In this context, we introduced a Schrödinger operator on M 1 , with potential determined by the volume of the fibers, and compared its spectrum with the spectrum of M 2 .…”