The metric measure boundary of spaces with Ricci curvature bounded below

Abstract

We solve a conjecture raised by Kapovitch, Lytchak, and Petrunin by showing that the metric measure boundary is vanishing on any $\mathrm{RCD}(K,N)$ 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.

References in corpus (9)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Semiconcave functions in Alexandrov's geometry
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• New differential operator and non-collapsed $RCD$ spaces
• Nonlinear diffusion equations and curvature conditions in metric measure spaces
• Metric-measure boundary and geodesic flow on Alexandrov spaces

Cited by in corpus (2)

• Symmetrized and non-symmetrized Asymptotic Mean Value Laplacian in metric measure spaces
• On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature