Ricci flow from singular spaces with bounded curvature

TL;DR

This work addresses constructing a Ricci flow starting from a singular, compact length space $(X,d)$ whose curvature is bounded above and below in the Alexandrov sense. By leveraging Hamilton–Cheeger–Gromov compactness, $C^{1,\alpha}$-regularity theory, and convergence results for limit flows, the authors obtain a smooth Ricci flow $(M,g(t))$, $t\in(0,T)$, whose initial GH-limit corresponds to $(X,d)$. They show that $g(t)$ converges to a $C^{1,\alpha}$-continuous metric $g$ on $M$ as $t\to0$ such that $(M,d_g)$ is isometric to $(X,d)$, and the flow is unique up to isometry. An equivariant version with group actions is also established, extending the result to symmetry constraints. Overall, the paper provides a rigorous smoothing mechanism for Alexandrov-type spaces under Ricci flow and clarifies the precise regularity and uniqueness properties of the resulting limit, with implications for understanding limits of manifolds under curvature bounds.

Abstract

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the $C^{1,α}$-sense to a $C^{1,α}$-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Remark 1.1
• Theorem 3
• Theorem 4
• Remark 1.2
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4