Hypersurfaces, Geodesics and Isoperimetric Inequalities in Cartan-Hadamard Manifolds

TL;DR

This paper extends classical isoperimetric inequalities to hypersurfaces in Cartan-Hadamard manifolds, relating submanifold geometry to geodesic measures and exploring spaces of geodesics in negatively curved spaces.

Contribution

It introduces new inequalities for submanifolds in non-positive curvature spaces and develops properties of geodesic spaces in Cartan-Hadamard manifolds.

Findings

Extension of Banchoff and Pohl's isoperimetric inequality

Modified Croke's inequality for hypersurfaces

Sharp isoperimetric inequality of Yau in negative curvature spaces

Abstract

We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of Banchoff and Pohl's isoperimetric inequality to spaces of non-positive curvature. We also prove a modified version of Croke's isoperimetric inequality for hypersurfaces immersed in Cartan-Hadamard manifolds and a sharp, quantitative version of an isoperimetric inequality of Yau in spaces of negative curvature. We discuss the relationship between these results, and we develop several facts about the spaces of geodesics in Cartan-Hadamards manifold that may be of independent interest.