The canon · Paper · Aug 21, 2020

Compactness theory of the space of Super Ricci flows

Richard H Bamler

math.DGarXiv:2008.09298v2
View PDF View HTML arXiv page

As on arXiv

Abstract

Bamler II. Compactness: sequences of super Ricci flows converge to synthetic metric flows.

We develop a compactness theory for super Ricci flows, which lays the foundations for the partial regularity theory in [Bam20b]. Our results imply that any sequence of super Ricci flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow. We will study the geometric and analytic properties of this limiting flow, as well as the convergence in detail. We will also see that, under appropriate local curvature bounds, a limit of Ricci flows can be decomposed into a regular and singular part. The regular part can be endowed with a canonical structure of a Ricci flow spacetime and we have smooth convergence on a certain subset of the regular part.

View PDF View HTML arXiv page
Submitted
Aug 21, 2020
Last revised
Jun 9, 2021 (v2)
Subjects
math.DG, math.AP, math.MG
DOI
10.1007/s00222-023-01196-3

Comments: 117 pages; minor corrections; Comments and questions welcome

Ricci flowWeak & singular flowsLimit spaces & RCD

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 Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)1Bamler 2020 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)7Bamler 2020 Synthetic notions of Ricci flow for metric measure spaces (Matthias Erbar, Zhenhao Li and Timo Schultz, 2025)4Erbar 2025 On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces (Xiang-Dong Li, 2025)5Li 2025 Synthetic approaches to Ricci flows (Matthias Erbar, Marco Flaim, Eric Hupp et al., 2025)3Erbar 2025 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)2Flaim 2026 On weak formulations of (super) Ricci flows (Sajjad Lakzian, 2026)6Lakzian 2026 Smoothing polyhedral spaces via Ricci flow (Richard H. Bamler and Esther Cabezas-Rivas, 2026)8Bamler 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Entropy and heat kernel bounds on a Ricci flow background

    Richard H BamlerThe canon

    Topics Limit spaces & RCD, Weak & singular flows · Same author Richard H Bamler · Shared terms “super Ricci”, “partial regularity”, “compactness”

  2. Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “super Ricci”, “synthetic”

  3. Synthetic approaches to Ricci flows

    Matthias Erbar, Marco Flaim, Eric Hupp et al.

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “synthetic”

  4. Synthetic notions of Ricci flow for metric measure spaces

    Matthias Erbar, Zhenhao Li and Timo Schultz

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “synthetic”

  5. On Perelman's -entropy and Shannon entropy power for super Ricci flows on metric measure spaces

    Xiang-Dong Li

    Topics Limit spaces & RCD, Weak & singular flows · Shared terms “super Ricci”

  6. On weak formulations of (super) Ricci flows

    Sajjad Lakzian

    Topics Weak & singular flows · Shared terms “super Ricci”

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

    Richard H BamlerThe canon

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

  8. Smoothing polyhedral spaces via Ricci flow

    Richard H. Bamler and Esther Cabezas-Rivas

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

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.

Foundational paper in the Weak & singular flows guide: Bamler: compactness theory for super Ricci flows and metric flows.

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) 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) 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)

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.