Volume growth of submanifolds and the Cheeger Isoperimetric Constant

TL;DR

This paper establishes a relationship between the volume growth of properly immersed submanifolds and the Cheeger isoperimetric constant within certain Riemannian manifolds, providing new geometric estimates.

Contribution

It provides a novel estimate of the Cheeger constant based on volume growth for submanifolds in manifolds with a pole and bounded sectional curvature.

Findings

Derived an upper bound for the Cheeger constant using volume growth

Connected geometric properties of submanifolds with curvature bounds

Extended previous results to a broader class of Riemannian manifolds

Abstract

We obtain an estimate of the Cheeger isoperimetric constant in terms of the volume growth for a properly immersed submanifold in a Riemannian manifold which possesses at least one pole and sectional curvature bounded from above .