Paper · Aug 27, 2026

Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

Kyeongho Bang, Eric Chen, Wenkui Du

math.DGarXiv:2608.26619
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Abstract

We establish the existence and uniqueness modulo time-independent diffeomorphisms of the positively curved ancient Ricci flow on a two-dimensional surface with boundary, assuming uniformly bounded diameter and constant positive boundary geodesic curvature. In particular, this ancient Ricci flow is rotationally symmetric, its backward limit is the flat disk, and its forward limit is a half-spherical singularity. To our knowledge, this result is the first instance of a classification result for ancient Ricci flows with boundary.

Ricci flowAncient solutionsSingularitiesSurfacesStability & uniquenessRotational symmetry & warped products

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Connected papers

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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)7Chan 2021 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)6Du 2022 Singular Ricci Flows on surfaces with boundary and positive scalar curvature (Jean C. Cortissoz and Juan J. Villamarín, 2023)5Cortissoz 2023 Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries (Fabio Podestà and Alberto Raffero, 2024)3Podestà 2024 A rigidity result for ancient Ricci flows (Qi S. Zhang, 2024)1Zhang 2024 Ancient Ricci flows with nonnegative Ricci curvature (Yuxing Deng, Ganqi Wang and Yongjia Zhang, 2026)8Deng 2026 Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere (Haixia Chen, Seunghyeok Kim and Monica Musso, 2026)4Chen 2026 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)2Daskalopoulo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A rigidity result for ancient Ricci flows

    Qi S. Zhang PDF

    Topics Ancient solutions, Singularities, Stability & uniqueness · Shared terms “backward limit”

  2. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al. PDF

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same author Wenkui Du · Shared terms “instance”

  3. Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries

    Fabio Podestà and Alberto Raffero PDF

    Topics Rotational symmetry & warped products · Shared terms “positively curved”, “rotationally”

  4. Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere

    Haixia Chen, Seunghyeok Kim and Monica Musso PDF

    Topics Rotational symmetry & warped products, Ancient solutions, Singularities · Shared terms “backward limit”, “rotationally symmetric”

  5. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín PDF

    Topics Surfaces, Rotational symmetry & warped products · Shared terms “geodesic curvature”

  6. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, Singularities, Stability & uniqueness · Same author Wenkui Du

  7. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “backward”

  8. Ancient Ricci flows with nonnegative Ricci curvature

    Yuxing Deng, Ganqi Wang and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities · Shared terms “positively”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.