Paper · Jun 28, 2020

Heat flow on time-dependent manifolds

Beomjun Choi, Jianhui Gao, Robert Haslhofer, Daniel Sigal

math.DGarXiv:2006.15678
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Abstract

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only depend on an upper bound for the logarithmic derivative of the volume measure. In particular, our estimates hold for any Ricci flow with scalar curvature bounded below, and such a lower bound of course depends only on the initial data.

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Submitted
Jun 28, 2020
Subjects
math.DG, math.AP

Comments: 12 pages

Ricci flowSingularitiesScalar curvatureWeak & singular flowsStability & uniqueness

Its neighbourhood

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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 Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows (Eva Kopfer and Karl-Theodor Sturm, 2016)3Kopfer 2016 Functional inequalities for the heat flow on time-dependent metric measure spaces (Eva Kopfer and Karl-Theodor Sturm, 2019)5Kopfer 2019 A note on the selfsimilarity of limit flows (Beomjun Choi, Robert Haslhofer and Or Hershkovits, 2019)8Choi 2019 Lectures on mean curvature flow of surfaces (Robert Haslhofer, 2021)6Haslhofer 2021 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)1Choi 2021 A note on blowup limits in 3d Ricci flow (Beomjun Choi and Robert Haslhofer, 2021)2Choi 2021 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)4Lee 2022 Gaussian Volume Functional, Integral Scalar Curvature, and Minimal Super-Ricci Flows (Marco Flaim, Erik Hupp and Karl-Theodor Sturm, 2026)7Flaim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Weak & singular flows, Singularities · Same authors Beomjun Choi, Robert Haslhofer

  2. A note on blowup limits in 3d Ricci flow

    Beomjun Choi and Robert Haslhofer

    Topics Weak & singular flows, Singularities · Same authors Beomjun Choi, Robert Haslhofer

  3. Heat Flows on Time-dependent Metric Measure Spaces and Super-Ricci Flows

    Eva Kopfer and Karl-Theodor Sturm

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “time dependent”, “heat flow”

  4. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Weak & singular flows, Scalar curvature, Stability & uniqueness · Shared terms “heat flow”, “lower bound”, “initial data”

  5. Functional inequalities for the heat flow on time-dependent metric measure spaces

    Eva Kopfer and Karl-Theodor Sturm

    Topics Weak & singular flows · Shared terms “time dependent”, “heat flow”, “logarithmic”

  6. Lectures on mean curvature flow of surfaces

    Robert Haslhofer

    Topics Weak & singular flows, Singularities · Same author Robert Haslhofer

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

    Marco Flaim, Erik Hupp and Karl-Theodor Sturm

    Topics Weak & singular flows, Scalar curvature · Shared terms “time dependent”, “Riemannian manifold”

  8. A note on the selfsimilarity of limit flows

    Beomjun Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Singularities · Same authors Beomjun Choi, Robert Haslhofer

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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)

In the guidebook

  1. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.