Extremal subsets in geodesically complete spaces with curvature bounded above

TL;DR

This paper introduces extremal subsets in geodesically complete spaces with curvature bounded above, exploring their properties and showing that topological singularities form such subsets under certain conditions.

Contribution

It defines extremal subsets in GCBA spaces and demonstrates their structural properties, extending concepts from Alexandrov spaces with curvature bounded below.

Findings

Topological singularities form extremal subsets under certain assumptions

Structural properties of extremal subsets in GCBA spaces are characterized

Extends the concept of extremal subsets to spaces with curvature bounded above

Abstract

We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analogue of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some structural properties of extremal subsets in GCBA spaces.