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For Riemannian submersions, we establish some estimates for the spectrum ofthe total space in terms of the spectrum of the base space and the geometry ofthe fibers. In particular, for Riemannian submersions of complete manifoldswith closed fibers of bounded mean curvature, we show that the spectrum of thebase space is discrete if and only if the spectrum of the total space isdiscrete.

Spectral estimates and discreteness of spectra under Riemannian submersions