Paper · Sep 29, 2026

Smoothing polyhedral spaces via Ricci flow

Richard H. Bamler, Esther Cabezas-Rivas

math.DGarXiv:2609.37549
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Abstract

We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.

Ricci flowLimit spaces & RCDStability & uniquenessHarmonic maps & DeTurck trick

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)6Bamler 2020 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)5Bamler 2020 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)7Huang 2021 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)8Lee 2022 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)4Bidabad 2023 Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds (Shouhei Honda, Christian Ketterer, Ilaria Mondello et al., 2023)3Honda 2023 Removing scalar curvature assumption for Ricci flow smoothing (Adam Martens, 2024)1Martens 2024 Ricci-harmonic flow of G₂ and Spin(7)-structures (Shubham Dwivedi, 2026)2Dwivedi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens PDF

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “smoothing”, “Gromov Hausdorff”, “short time”

  2. Ricci-harmonic flow of and Spin(7)-structures

    Shubham Dwivedi PDF

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “short time”, “time existence”

  3. Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

    Shouhei Honda, Christian Ketterer, Ilaria Mondello et al. PDF

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “Gromov Hausdorff”, “curvature bounds”, “Euclidean”

  4. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  5. Structure theory of non-collapsed limits of Ricci flows

    Richard H BamlerThe canon PDF

    Topics Limit spaces & RCD · Same author Richard H. Bamler · Shared terms “curvature bounds”

  6. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon PDF

    Topics Limit spaces & RCD · Same author Richard H. Bamler · Shared terms “volume bound”, “pointwise”, “curvature bounds”

  7. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  8. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping PDF

    Topics Limit spaces & RCD · Shared terms “positive time”, “Hausdorff limit”, “curvature estimates”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.