Paper · Feb 20, 2024

Scalar curvature rigidity of the four-dimensional sphere

Simone Cecchini, Jinmin Wang, Zhizhang Xie, Bo Zhu

math.DGarXiv:2402.12633v2
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Abstract

Let be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by . In this paper, we prove that if is a smooth map of non-zero degree from to the unit four-sphere, then is an isometry. Following ideas of Gromov, we use -bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.

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Submitted
Feb 20, 2024
Last revised
Mar 20, 2024 (v2)
Subjects
math.DG

Comments: Improved exposition. Comments are welcome!

Ricci flow4-manifoldsScalar curvatureStability & uniquenessHarmonic maps & DeTurck trick

Its neighbourhood

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  1. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “smooth map”, “zero degree”

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

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “map heat”, “heat flow”, “flow coupled”

  3. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon

    Topics Harmonic maps & DeTurck trick, 4-manifolds, Scalar curvature · Shared terms “possibly”

  4. Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

    Pinsard Morel Lucas

    Topics Harmonic maps & DeTurck trick, 4-manifolds · Shared terms “four dimensional”

  5. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics 4-manifolds, Scalar curvature · Shared terms “four dimensional”, “Let”

  6. Rigidity of four-dimensional Kähler-Ricci solitons

    Xiaodong Cao, Ernani Ribeiro and Hung Tran

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “four dimensional”

  7. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Scalar curvature, Stability & uniqueness · Shared terms “curvature rigidity”

  8. Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

    Shouhei Honda, Christian Ketterer, Ilaria Mondello et al.

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “map heat”, “Gromov”, “heat flow”

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 Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 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) 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. 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.