The canon · Paper · Feb 25, 2021

Recent developments in Ricci flows

Richard H Bamler

math.DGarXiv:2102.12615
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Abstract

Bamler. A survey of recent developments — a map of the modern field.

This is a survey on recent developments in Ricci flows.

Comments: 16 pages

Ricci flow

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 Uniqueness and stability of Ricci flow through singularities (Richard H. Bamler and Bruce Kleiner, 2017)5Bamler 2017 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)4Bamler 2020 On the fundamental group of non-collapsed ancient Ricci flows (Richard H. Bamler, 2021)2Bamler 2021 Singularity models in the three-dimensional Ricci flow (S. Brendle, 2022)8Brendle 2022 Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments (Karl-Theodor Sturm, 2024)7Sturm 2024 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)1Tosatti 2024 Smoothing polyhedral spaces via Ricci flow (Richard H. Bamler and Esther Cabezas-Rivas, 2026)3Bamler 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti PDF

    Topics Ricci flow

  2. On the fundamental group of non-collapsed ancient Ricci flows

    Richard H. Bamler PDF

    Topics Ricci flow · Same author Richard H Bamler

  3. Smoothing polyhedral spaces via Ricci flow

    Richard H. Bamler and Esther Cabezas-Rivas PDF

    Topics Ricci flow · Same author Richard H Bamler

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

    Richard H BamlerThe canon PDF

    Topics Ricci flow · Same author Richard H Bamler

  5. Uniqueness and stability of Ricci flow through singularities

    Richard H. Bamler and Bruce KleinerThe canon PDF

    Topics Ricci flow · Same author Richard H Bamler

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

    Richard H BamlerThe canon PDF

    Topics Ricci flow · Same author Richard H Bamler

  7. Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

    Karl-Theodor Sturm PDF

    Topics Ricci flow

  8. Singularity models in the three-dimensional Ricci flow

    S. Brendle PDF

    Topics Ricci flow

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.

Survey in the Ancient solutions guide: Bamler's survey, which places the classification in the modern theory.

Survey in the Smale conjecture & diffeomorphisms guide: Bamler's survey, with an accessible account of these applications.

Survey in the Heat kernel guide: Bamler's survey, which explains the heat-kernel approach informally.

Survey in the Limit spaces & RCD guide: Bamler's survey, comparing Ricci flow limits with Cheeger–Colding theory.

Survey in the Ricci flow guide: Bamler's survey of recent developments: the map of the modern field.

Survey in the Singularities guide: Bamler's survey of recent developments.

Survey in the Weak & singular flows guide: Bamler's survey, including the metric-flow framework.

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)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.