Paper · Mar 13, 2016

Structure theory of singular spaces

Richard H. Bamler

math.DGarXiv:1603.05236v2
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Abstract

In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded curvature, which we will describe in a subsequent paper.

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Submitted
Mar 13, 2016
Last revised
Jul 7, 2016 (v2)
Subjects
math.DG, math.AP

Comments: 105 pages, minor updates, formerly part of arXiv:1512.08527

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 2016, 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 Convergence of Ricci flows with bounded scalar curvature (Richard H. Bamler, 2016)5Bamler 2016 The Ricci flow under almost non-negative curvature conditions (Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking, 2017)8Bamler 2017 Entropy and heat kernel bounds on a Ricci flow background (Richard H Bamler, 2020)4Bamler 2020 Compactness theory of the space of Super Ricci flows (Richard H Bamler, 2020)1Bamler 2020 Structure theory of non-collapsed limits of Ricci flows (Richard H Bamler, 2020)3Bamler 2020 Convergence of the Yamabe flow on singular spaces with positive Yamabe constant (Gilles Carron, Jørgen Olsen Lye and Boris Vertman, 2021)7Carron 2021 Ricci flow from spaces with edge type conical singularities (Lucas Lavoyer, 2023)6Lavoyer 2023 Ricci flow from singular spaces with bounded curvature (Diego Corro, Masoumeh Zarei and Adam Moreno, 2025)2Corro 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compactness theory of the space of Super Ricci flows

    Richard H Bamler

    Topics Limit spaces & RCD, Weak & singular flows · Same author Richard H. Bamler · Shared terms “sequence”

  2. Ricci flow from singular spaces with bounded curvature

    Diego Corro, Masoumeh Zarei and Adam Moreno

    Topics Weak & singular flows · Shared terms “singular spaces”, “bounded curvature”, “Riemannian manifold”

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

    Richard H Bamler

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

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

    Richard H Bamler

    Topics Limit spaces & RCD, Weak & singular flows · Same author Richard H. Bamler · Shared terms “structure theory”, “Naber”

  5. Convergence of Ricci flows with bounded scalar curvature

    Richard H. Bamler

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

  6. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer

    Topics Limit spaces & RCD, Weak & singular flows

  7. Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

    Gilles Carron, Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Limit spaces & RCD, Weak & singular flows

  8. The Ricci flow under almost non-negative curvature conditions

    Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking

    Topics Weak & singular flows · Same author Richard H. Bamler · Shared terms “singular spaces”, “sequence”

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) 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.