Wall singularity of spaces with an upper curvature bound

TL;DR

This paper investigates the structure of wall singularities in metric spaces with an upper curvature bound, providing geometric characterizations and conditions for the codimension of singular sets.

Contribution

It offers a geometric structure theorem for codimension one singularities and characterizes codimension two regularity in such spaces.

Findings

Provides necessary and sufficient conditions for singular sets to have codimension at least two.

Characterizes wall singularities of codimension one in metric spaces with curvature bounds.

Establishes a geometric framework for understanding singularities in these spaces.

Abstract

We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.