Gluing of multiple Alexandrov spaces

Abstract

Perel'man's Doubling Theorem [11] says that the doubling of an Alexandrov space is also an Alexandrov space with the same lower curvature bound. Petrunin [12] generalizes this theorem to two Alexandrov spaces being glued along their isometric boundaries. These theorems have a lot of applications to the study of spaces with lower curvature bounds. As a generalization to these results and a tool to construct and verify new Alexandrov spaces, we study the gluing of multiple Alexandrov spaces. We consider a Gluing Conjecture, which says that the gluing of finite number of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to be Alexandrov spaces.We prove the Gluing Conjecture when the complement of (n−1,ϵ)-regular points is discrete in the glued part. In particular, this implies the Gluing Conjecture in dimension 2. This is a new result in the case of gluing 2-dimensional convex sets. In our proof, we also establish some structural theory for the general gluing. Based on the tangent cone condition, the 2-point gluing over (n−1,ϵ)-regular points is classified to be locally separable, as being assumed in Petrunin's Gluing Theorem. The gluing near an un-glued (n−1,ϵ)-regular point is classified to be involutional.