Isometric immersions of RCD$(K,N)$ spaces via heat kernels

TL;DR

This paper proves that compact RCD$(K,N)$ spaces that can be isometrically immersed via heat kernels are actually smooth Riemannian manifolds, establishing a deep link between heat kernel properties and manifold regularity.

Contribution

It shows that isometric heat kernel immersions characterize smooth Riemannian manifolds within RCD spaces, extending previous regularity results and providing new compactness theorems.

Findings

Compact RCD$(K,N)$ spaces with heat kernel isometric immersions are smooth manifolds.

Any such space is isometric to an unweighted closed Riemannian manifold.

The results lead to a $C^ abla$-compactness theorem for certain Riemannian manifolds.

Abstract

Given an RCD$(K,N)$ space $({X},\mathsf{d},\mathfrak{m})$, one can use its heat kernel $\rho$ to map it into the $L^2$ space by a locally Lipschitz map $\Phi_t(x):=\rho(x,\cdot,t)$. The space $(X,\mathsf{d},\mathfrak{m})$ is said to be an isometrically heat kernel immersing space, if each $\Phi_t$ is an isometric immersion {}{after a normalization}. A main result states that any compact isometrically heat kernel immersing RCD$(K,N)$ space is isometric to an unweighted closed smooth Riemannian manifold. This is justified by a more general result: if a compact non-collapsed RCD$(K, N)$ space has an isometrically immersing eigenmap, then the space is isometric to an unweighted closed Riemannian manifold, which greatly improves a regularity result in \cite{H21} by Honda. As an application of these results, we give a $C^\infty$-compactness theorem for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an isometrically immersing eigenmap.