Asymptotic geometric regularity of CAT(0) spaces

TL;DR

This paper demonstrates that certain high-dimensional CAT(0) spaces with boundaries close to a sphere are bi-Lipschitz homeomorphic to Euclidean space, revealing geometric regularity under boundary conditions.

Contribution

It establishes a new link between boundary geometry and the global structure of CAT(0) spaces, extending understanding of their asymptotic regularity.

Findings

CAT(0) spaces with boundary close to a sphere are bi-Lipschitz homeomorphic to Euclidean space

Similar bi-Lipschitz homeomorphism results hold for CAT(1) spaces close to a sphere

Provides conditions under which geometric regularity emerges in non-positively curved spaces

Abstract

We prove that if an n-dimensional geodesically complete CAT(0) space has Tits boundary sufficiently close to the (n-1)-dimensional standard unit sphere, then it is bi-Lipschiz homeomorphic to the n-dimensional Euclidean space. As an application, we conclude that if an (n-1)-dimensional geodesically complete CAT(1) space is sufficiently close to the (n-1)-dimensional standard unit sphere, then they are bi-Lipschiz homeomorphic to each other.