Self and partial gluing theorems for Alexandrov spaces with a lower curvature bound

TL;DR

This paper proves that gluing along extremal subsets with isometric involutions in Alexandrov spaces preserves the lower curvature bound, generalizing Perelman's doubling and Petrunin's gluing theorems.

Contribution

It introduces new gluing theorems for Alexandrov spaces involving extremal subsets and isometric involutions, extending existing results.

Findings

Gluing along extremal subsets preserves curvature bounds.

Generalization of Perelman's doubling theorem.

Extension of Petrunin's gluing theorem.

Abstract

This paper is devoted to prove that if an Alexandrov space of curvature not less than $\kappa$ with a codimension one extremal subset which admits an isometric involution with respect to the induced length metric, then the metric space obtained by gluing the extremal subset along the isometry is an Alexandrov space of curvature not less than $\kappa$. This is a generalization of Perelman's doubling and Petrunin's gluing theorems.