Paper · Dec 18, 2023

Rigidity of area non-increasing maps

Man-Chun Lee, Luen-Fai Tam, Jingbo Wan

math.DGarXiv:2312.10940v2
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Abstract

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between certain positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This implies that an area non-increasing self map of , is either homotopically trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow to establish related results on manifolds with positive -isotropic curvature.

Comments: 35 pages, minor changes in abstract and introduction, all comments are welcome

Ricci flowMean curvature flowSphere theoremsStability & uniqueness

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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 Manifolds with 1/4-pinched Curvature are Space Forms (S. Brendle and R. M. Schoen, 2007)8Brendle 2007 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)1Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)3Lee 2022 Ricci flow and PIC1 (Peter M. Topping, 2023)6Topping 2023 On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces (Nurlan Abiev, 2024)4Abiev 2024 Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen (David González-Álvaro and Masoumeh Zarei, 2024)7González-Álv 2024 Expanding Ricci solitons coming out of weakly PIC1 metric cones (Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey, 2024)5Chan 2024 Small Normal Curvature and Three-Manifold Topology (Tsz-Kiu Aaron Chow and Jingbo Wan, 2026)2Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Stability & uniqueness · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “flow coupled”, “isometry”

  2. Small Normal Curvature and Three-Manifold Topology

    Tsz-Kiu Aaron Chow and Jingbo Wan PDF

    Topics Sphere theorems · Same author Jingbo Wan · Shared terms “immersion”, “Brendle”

  3. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Sphere theorems · Same author Man-Chun Lee

  4. On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces

    Nurlan Abiev PDF

    Topics Sphere theorems · Shared terms “positively curved”

  5. Expanding Ricci solitons coming out of weakly PIC1 metric cones

    Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey PDF

    Topics Sphere theorems · Same author Man-Chun Lee

  6. Ricci flow and PIC1

    Peter M. Topping PDF

    Topics Sphere theorems · Shared terms “positive curvature”

  7. Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen

    David González-Álvaro and Masoumeh Zarei PDF

    Topics Sphere theorems · Shared terms “positively curved”

  8. Manifolds with 1/4-pinched Curvature are Space Forms

    S. Brendle and R. M. SchoenThe canon PDF

    Topics Sphere theorems · Shared terms “isotropic curvature”

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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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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)

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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  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.