Paper · Jul 29, 2020

Defining Pointwise Lower Scalar Curvature Bounds for Metrics with Regularization by Ricci Flow

Paula Burkhardt-Guim

math.DGarXiv:2007.14967v3
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Abstract

We survey some recent work using Ricci flow to create a class of local definitions of weak lower scalar curvature bounds that is well defined for metrics. We discuss several properties of these definitions and explain some applications of this approach to questions regarding uniform convergence of metrics with scalar curvature bounded below. Finally, we consider the relationship between this approach and some other generalized notions of lower scalar curvature bounds.

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Submitted
Jul 29, 2020
Last revised
Dec 17, 2020 (v3)
Subjects
math.DG
Journal
SIGMA 16 (2020), 128, 10 pages
DOI
10.3842/SIGMA.2020.128

Comments: This paper surveys work on the generalized definition of lower scalar curvature bounds for metrics given in arXiv:1907.13116, and states some more streamlined results

Ricci flowScalar curvature

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 Pointwise lower scalar curvature bounds for C⁰ metrics via regularizing Ricci flow (Paula Burkhardt-Guim, 2019)1Burkhardt-Gu 2019 Global ε-regularity for 4-dimensional Ricci flow with integral scalar curvature bound (Wangjian Jian, 2020)6Jian 2020 On scalar curvature lower bounds and scalar curvature measure (John Lott, 2021)2Lott 2021 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)8Jiang 2021 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)5Burkhardt-Gu 2022 A continuous Positive Mass Theorem for perturbations of Euclidean space (Paula Burkhardt-Guim, 2026)4Burkhardt-Gu 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)3Huang 2026 Scalar curvature bounds and rigidity of Ricci shrinkers (Jianyu Ou and Guoqiang Wu, 2026)7Ou 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pointwise lower scalar curvature bounds for metrics via regularizing Ricci flow

    Paula Burkhardt-Guim

    Topics Scalar curvature · Same author Paula Burkhardt-Guim · Shared terms “lower scalar”, “pointwise lower”, “curvature bounds”

  2. On scalar curvature lower bounds and scalar curvature measure

    John Lott

    Topics Scalar curvature · Shared terms “lower scalar”, “curvature bounds”

  3. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang

    Topics Scalar curvature · Shared terms “curvature bounds”, “pointwise”

  4. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim

    Topics Scalar curvature · Same author Paula Burkhardt-Guim

  5. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim

    Topics Scalar curvature · Same author Paula Burkhardt-Guim

  6. Global -regularity for 4-dimensional Ricci flow with integral scalar curvature bound

    Wangjian Jian

    Topics Scalar curvature · Shared terms “curvature bounds”

  7. Scalar curvature bounds and rigidity of Ricci shrinkers

    Jianyu Ou and Guoqiang Wu

    Topics Scalar curvature · Shared terms “curvature bounds”, “pointwise”

  8. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang

    Topics Scalar curvature

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 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.