Asymptotic geometric regularity of CAT(0) 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.

21 pages. arXiv admin note: substantial text overlap with arXiv:2601.22673