Paper · May 14, 2026

-solutions with the round cylinder as an asymptotic shrinker

Aprameya Girish Hebbar

math.DGarXiv:2605.14905
PDF Abstract page

As on arXiv

Abstract

We show that -solutions to the Ricci flow in dimensions whose asymptotic shrinking Ricci soliton is the round cylinder must be uniformly PIC. Combined with earlier classification results, this implies that any such noncompact solution is either the round shrinking cylinder or the Bryant steady soliton, and any such compact solution is Perelman's ancient solution.

Ricci flowRicci solitonsAncient solutionsSphere theoremsBoundaries & non-compact

Its neighbourhood

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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)2Brendle 2018 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)3Li 2021 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)4Lynch 2022 Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature (Huai-Dong Cao and Junming Xie, 2024)7Cao 2024 On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions (Zhengnan Chen, 2025)5Chen 2025 Asymptotic Geometry of Four-Dimensional Steady Solitons (Aprameya Girish Hebbar and Natasa Sesum, 2026)1Hebbar 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)6Daskalopoulo 2026 Ricci flow on manifolds with positive isotropic curvature in all dimensions (Simon Brendle and Raphael Tsiamis, 2026)8Brendle 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic Geometry of Four-Dimensional Steady Solitons

    Aprameya Girish Hebbar and Natasa Sesum PDF

    Topics Ancient solutions, Boundaries & non-compact, Ricci solitons · Same author Aprameya Girish Hebbar · Shared terms “steady solitons”, “Bryant”

  2. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon PDF

    Topics Ancient solutions, Boundaries & non-compact · Shared terms “shrinking cylinder”, “Bryant”, “Perelman”

  3. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang PDF

    Topics Ricci solitons · Shared terms “round cylinder”

  4. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego PDF

    Topics Sphere theorems, Ancient solutions, Boundaries & non-compact · Shared terms “uniformly PIC”

  5. On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

    Zhengnan Chen PDF

    Topics Sphere theorems, Ancient solutions, Ricci solitons · Shared terms “uniformly PIC”, “shrinking Ricci”

  6. Unique asymptotics of symmetric ancient ovals of Ricci flow

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

    Topics Sphere theorems, Ancient solutions · Shared terms “Perelman ancient”

  7. Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

    Huai-Dong Cao and Junming Xie PDF

    Topics Sphere theorems, Ancient solutions, Ricci solitons · Shared terms “shrinking Ricci”, “steady solitons”, “PIC”

  8. Ricci flow on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis PDF

    Topics Sphere theorems, Ancient solutions, Boundaries & non-compact · Shared terms “shrinking Ricci”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.