Paper · Oct 2, 2026

On the uniqueness of Ricci flows from certain Alexandrov spaces

Laura Flower

math.DGarXiv:2610.03129
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Abstract

We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon–Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.

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Submitted
Oct 2, 2026
Subjects
math.DG
Ricci flow(Non)collapsingWeak & singular flowsStability & uniquenessBoundaries & non-compact

Its neighbourhood

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    Topics Weak & singular flows · Shared terms “initial data”, “curvature lower”, “lower bound”

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    Topics (Non)collapsing, Weak & singular flows, Boundaries & non-compact · Shared terms “Reifenberg”, “PIC”

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    Hongting Ding, Shaochuang Huang and Zhuo Peng

    Topics (Non)collapsing, Boundaries & non-compact · Shared terms “invariant curvature”, “curvature lower”, “complete noncompact”

  6. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

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    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “Lai”, “sectional curvature”, “complete noncompact”

  7. Rotationally symmetric Ricci Flow on

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    Topics (Non)collapsing, Weak & singular flows · Shared terms “scaling invariant”, “invariant curvature”, “curvature bounds”

  8. On the structure of noncollapsed Ricci flow limit spaces

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    Topics (Non)collapsing · Shared terms “limit space”, “noncollapsed”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.