On strictly convex subsets in negatively curved manifolds

TL;DR

This paper establishes a precise criterion based on extrinsic curvature for when a subset in a negatively curved manifold is an epsilon-neighborhood of a convex set, enhancing understanding of convexity in such geometries.

Contribution

It provides a sharp, curvature-based criterion characterizing epsilon-neighborhoods of convex subsets in negatively curved manifolds.

Findings

Sharp curvature criterion for convex neighborhoods

Characterization of epsilon-neighborhoods in negatively curved spaces

Advances understanding of convexity in Riemannian geometry

Abstract

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.