Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds

TL;DR

The paper addresses when harmonic RCD$(K,N)$ spaces are smooth by extending classical harmonic manifold theory to non-smooth synthetic spaces. It introduces strongly harmonic and radially symmetric RCD spaces, proves measure rigidity and smoothness in several regimes, and analyzes volume-homogeneous spaces via co-area and disintegration techniques to obtain bi-Lipschitz coordinates and ultimately smooth Riemannian structures. The main contributions include showing compact strongly harmonic radially symmetric RCD spaces are smooth, and establishing smoothness under volume-homogeneity by reducing to a non-collapsed, Euclidean-like setting with radial symmetry. These results bridge synthetic metric-measure geometry with classical Riemannian regularity, providing a synthetic analogue to harmonic manifolds and strengthening the link between heat kernel structure, volume growth, and smoothness.

Abstract

In this paper, we study harmonic RCD$(K,N)$ spaces as the counterpart of harmonic Riemannian manifolds with Ricci curvature bounded from below. We prove that a compact RCD$(K,N)$ space is isometric to a smooth closed Riemannian manifold if it satisfies either of the following harmonicity conditions:(1) the heat kernel $ρ(x,y,t)$ depends only on the variable $t$ and the distance between points $x$ and $y$; (2) the volume of the intersection of two geodesic balls depends only on their radii and the distance between their centers.

Theorems & Definitions (76)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: CH11, CH12
• Theorem 1.4: S90
• Theorem 1.5: S90
• Definition 1.6: Strongly harmonic RCD$(K,N)$ space
• Definition 1.7: Radially symmetric RCD$(K,N)$ space
• Theorem 1.8: Measure structure of strongly harmonic spaces
• Theorem 1.9: Smoothness of radially symmetric spaces
• Corollary 1.10