The metric measure boundary of spaces with Ricci curvature bounded below

TL;DR

This paper proves that the metric measure boundary vanishes in ${ m RCD}(K,N)$ spaces without boundary, resolving a conjecture and an open question about geodesics in Alexandrov spaces.

Contribution

It confirms the vanishing of the metric measure boundary in ${ m RCD}(K,N)$ spaces without boundary, settling a longstanding conjecture and related open problem.

Findings

Metric measure boundary vanishes in ${ m RCD}(K,N)$ spaces without boundary

Resolved a conjecture by Kapovitch, Lytchak, and Petrunin

Answered an open question about infinite geodesics in Alexandrov spaces

Abstract

We solve a conjecture raised by Kapovitch, Lytchak, and Petrunin by showing that the metric measure boundary is vanishing on any ${\rm RCD}(K,N)$ space without boundary. Our result, combined with [Kapovitch-Lytchak-Petrunin '21], settles an open question about the existence of infinite geodesics on Alexandrov spaces without boundary raised by Perelman and Petrunin in 1996.