On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds

TL;DR

This paper establishes conditions under which the topological boundary of certain subspaces in metric measure spaces with lower curvature bounds coincides with their intrinsic boundaries, linking geometric and topological properties.

Contribution

It proves boundary coincidence results for Alexandrov and RCD spaces, extending understanding of boundary behavior in spaces with lower curvature bounds.

Findings

Topological boundary coincides with Alexandrov boundary in Alexandrov spaces.

Topological boundary coincides with De Philippis-Gigli boundary in RCD spaces.

Results imply convexity properties of boundary structures.

Abstract

We show that if an Alexandrov space $X$ has an Alexandrov subspace $\bar \Omega$ of the same dimension disjoint from the boundary of $X$, then the topological boundary of $\bar \Omega$ coincides with its Alexandrov boundary. Similarly, if a noncollapsed RCD(K,N) space $X$ has a noncollapsed RCD(K,N) subspace $\bar \Omega$ disjoint from boundary of $X$ and with mild boundary condition, then the topological boundary of $\bar \Omega$ coincides with its De Philippis-Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.