Improvements of upper curvature bounds

TL;DR

This paper demonstrates that spaces with positive upper curvature bounds can be locally modified to have closely related regions with negative upper curvature bounds, revealing new local geometric flexibility.

Contribution

It introduces a method to locally alter the curvature bounds of a space, connecting positive and negative upper curvature bounds in small neighborhoods.

Findings

Spaces with positive upper curvature bounds can be locally approximated by regions with negative upper curvature bounds.

The local curvature bounds can be closely related, enabling new geometric constructions.

This provides insights into the local structure and flexibility of curvature bounds in metric spaces.

Abstract

We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.