Paper · Sep 16, 2018

A conformally invariant gap theorem characterizing via the Ricci flow

Sun-Yung A. Chang, Matthew Gursky, Siyi Zhang

math.DGarXiv:1809.05918
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Abstract

We extend the sphere theorem of to give a conformally invariant characterization of . In particular, we introduce a conformal invariant defined on conformal four-manifolds satisfying a `positivity' condition; it follows from that if , then is diffeomorphic to . Our main result of this paper is a `gap' result showing that if and for small enough, then is diffeomorphic to . The Ricci flow is used in a crucial way to pass from the bounds on to pointwise curvature information.

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Submitted
Sep 16, 2018
Subjects
math.DG

Comments: 26 pages

Ricci flow4-manifoldsSphere theorems

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  1. Some aspects of Ricci flow on the 4-sphere

    Sun-Yung Alice Chang and Eric Chen

    Topics Ricci flow · Same author Sun-Yung A. Chang · Shared terms “gap theorem”, “conformal invariant”

  2. Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

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    Topics Sphere theorems, 4-manifolds

  3. Ricci flow does not preserve positive sectional curvature in dimension four

    Renato G. Bettiol and Anusha M. Krishnan

    Topics Sphere theorems, 4-manifolds

  4. Four-Dimensional Steady Gradient Ricci Solitons with -Cylindrical Tangent Flows at Infinity

    Richard Bamler, Bennett Chow, Yuxing Deng et al.

    Topics Sphere theorems, 4-manifolds

  5. -symmetry of four-dimensional steady Ricci solitons

    Hanbing Fang

    Topics Sphere theorems, 4-manifolds · Shared terms “diffeomorphic”

  6. Curvature decompositions on Einstein four-manifolds

    Peng Wu

    Topics 4-manifolds · Shared terms “four manifolds”, “positivity condition”

  7. Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

    Huai-Dong Cao and Junming Xie

    Topics Sphere theorems, 4-manifolds · Shared terms “characterization”

  8. Quasiconformal Flows on non-Conformally Flat Spheres

    Sun-Yung Alice Chang, Eden Prywes and Paul Yang

    Topics Ricci flow · Same author Sun-Yung A. Chang · Shared terms “conformally”

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 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)

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.