Topology and bottom spectrum of transversally negatively curved foliations

TL;DR

This paper investigates the geometric and spectral properties of Riemannian foliations with negatively curved leaf spaces, establishing diffeomorphism of the normal exponential map and providing sharp spectral estimates.

Contribution

It proves the normal exponential map is a diffeomorphism under negative curvature and derives optimal spectral bounds for minimal leaves and horizontal Laplacians.

Findings

Normal exponential map is a diffeomorphism for negatively curved leaf spaces.

Provides sharp estimates for the bottom of the spectrum of the manifold.

Establishes spectral bounds for the horizontal Laplacian.

Abstract

We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a sharp estimate for the bottom of the spectrum of such a Riemannian manifold. Our proof of the spectral estimate also yields an estimate for the bottom of the spectrum of the horizontal Laplacian.