Paper · Dec 8, 2020

Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric

Tsz-Kiu Aaron Chow

math.DGarXiv:2012.04430v2
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Abstract

In this paper, we study the Ricci flow on manifolds with boundary. In the first part of the paper, we prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously umbilic for positive time. In the second part of the paper, we prove that the flow we constructed in the first part preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

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Submitted
Dec 8, 2020
Last revised
Aug 7, 2021 (v2)
Subjects
math.DG, math.AP

Comments: 55 pages, improved exposition

Ricci flowSphere theoremsStability & uniquenessBoundaries & non-compactHarmonic maps & DeTurck trick

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 Manifolds with positive curvature operators are space forms (Christoph Boehm and Burkhard Wilking, 2006)7Boehm 2006 Weak stability of Ricci expanders with positive curvature operator (Alix Deruelle and Tobias Lamm, 2016)8Deruelle 2016 Weakly PIC1 manifolds with maximal volume growth (Fei He and Man-Chun Lee, 2018)3He 2018 Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci (Mahdi Ammar, 2019)5Ammar 2019 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)6Huang 2021 Ricci flow and PIC1 (Peter M. Topping, 2023)1Topping 2023 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)4Chow 2026 Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres (Jason DeVito, David González-Álvaro and Masoumeh Zarei, 2026)2DeVito 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Shared terms “PIC”, “positive curvature”

  2. Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres

    Jason DeVito, David González-Álvaro and Masoumeh Zarei

    Topics Sphere theorems · Shared terms “preserves positive”, “flow preserves”

  3. Weakly PIC1 manifolds with maximal volume growth

    Fei He and Man-Chun Lee

    Topics Sphere theorems · Shared terms “PIC condition”

  4. A Spinorial Perelman's Functional: Critical Points and Gradient Flow

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same author Tsz-Kiu Aaron Chow · Shared terms “short time”, “time existence”

  5. Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci

    Mahdi Ammar

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “initial metric”

  6. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact, Stability & uniqueness · Shared terms “short time”, “time existence”

  7. Manifolds with positive curvature operators are space forms

    Christoph Boehm and Burkhard WilkingThe canon

    Topics Sphere theorems · Shared terms “positive curvature”, “curvature operator”

  8. Weak stability of Ricci expanders with positive curvature operator

    Alix Deruelle and Tobias Lamm

    Topics Sphere theorems, Stability & uniqueness · Shared terms “positive curvature”, “curvature operator”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 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.

  3. 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.