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arXiv · 2412.20841

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

Zhangkai Huang
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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 $\rho(x,y,t)$ 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.

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Zhangkai Huang. 2024-12-30. Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds. https://doi.org/10.1016/j.na.2026.114269

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