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.
Highlights
The spectrum of the Laplacian on a Riemannian manifold is an isometric invariant whose relation with the geometry of the manifold is not comprehended completely
Its behavior under maps between Riemannian manifolds, which respect the geometry of the manifolds to some extent, remains largely unclear
We study the behavior of the spectrum under Riemannian submersions
Summary
The spectrum of the Laplacian on a Riemannian manifold is an isometric invariant whose relation with the geometry of the manifold is not comprehended completely. If the submersion has fibers of basic mean curvature, we prove that the (essential) spectrum of S is contained in the (essential, respectively) spectrum of the Laplacian on M2 This is formulated in the following generalization of [2, Theorem 1]. If the submersion has closed fibers of bounded mean curvature, it is easy to estimate the bottom of the (essential) spectrum of S in terms of the bottom of the (essential, respectively) spectrum of the Laplacian on M1 and the mean curvature of the fibers In this setting, we obtain the following application of Theorem 1.2.
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