On the fundamental tone of immersions and submersions

Abstract

Abstract. In this paper we obtain lower bound estimates of the spectrum of Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previous known estimates and applies for both immersions and submersions.

Spectral estimates and discreteness of spectra under Riemannian submersions

“…In the first part of the paper, we establish a lower bound for the bottom of the spectrum of the total space, under the assumption that the (unnormalized) mean curvature H of the fibers is bounded in a specific way. In particular, we extend the recent result of [5] about Riemannian submersions. According to [5,Theorem 1.1], if M 1 is the m-dimensional hyperbolic space H m , and the mean curvature vector field of the fibers is bounded by H ≤ C ≤ m − 1, then the bottom of the spectrum of the Laplacian on M 2 satisfies…”

“…It is worth to point out that a similar estimate (without the last term) may be derived from [5,Theorem 5.1]. However, Theorem 1.1 yields a sharper estimate than [5, Theorem 5.1] for submersions over negatively curved symmetric spaces.…”

Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature

“…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…”

Rigidity and stability estimates for minimal submanifolds in the hyperbolic space

“…Simon's work has been of great interest to differential geometers, and in the last decade several interesting gap theorems for submanifolds have been successfully obtained. We refer the reader to [1], [2], [3], [4], [5], [7], [9], [10], [11], [13], [14], and the references therein.…”