Paper · Mar 28, 2019

Curvature decompositions on Einstein four-manifolds

Peng Wu

math.DGarXiv:1903.11817
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Abstract

For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest possible new ideas to study the well-known rigidity conjecture for positively curved Einstein four-manifolds.

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Submitted
Mar 28, 2019
Subjects
math.DG
Journal
New York J. Math. 23 (2017), 1739-1749

Comments: This is a part of the author's Ph.D. thesis at University of California, Santa Barbara in 2012

Ricci flow4-manifoldsScalar curvatureStability & uniqueness

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 Rigidity of positively curved shrinking Ricci solitons in dimension four (Giovanni Catino, 2018)1Catino 2018 Four-dimensional complete gradient shrinking Ricci solitons (Huai-Dong Cao, Ernani Ribeiro and Detang Zhou, 2020)5Cao 2020 Global ε-regularity for 4-dimensional Ricci flow with integral scalar curvature bound (Wangjian Jian, 2020)8Jian 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)3Cheng 2021 Curvature estimates for four-dimensional complete gradient expanding Ricci solitons (Huai-Dong Cao and Tianbo Liu, 2021)7Cao 2021 Scalar curvature rigidity of the four-dimensional sphere (Simone Cecchini, Jinmin Wang, Zhizhang Xie et al., 2024)6Cecchini 2024 Rigidity of compact four-dimensional weakly Einstein Ricci Solitons (JeongHyeong Park and Wooseok Shin, 2026)2Park 2026 L² Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (Xianbin Huang and Wancheng Zhang, 2026)4Huang 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 positively curved shrinking Ricci solitons in dimension four

    Giovanni Catino

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness · Shared terms “positively curved”

  2. Rigidity of compact four-dimensional weakly Einstein Ricci Solitons

    JeongHyeong Park and Wooseok Shin

    Topics 4-manifolds, Stability & uniqueness · Shared terms “Einstein four”, “four manifolds”, “relation”

  3. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness

  4. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang

    Topics 4-manifolds, Scalar curvature · Shared terms “decomposition”

  5. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics 4-manifolds, Scalar curvature

  6. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al.

    Topics 4-manifolds, Scalar curvature, Stability & uniqueness

  7. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu

    Topics 4-manifolds, Scalar curvature · Shared terms “curvature tensor”

  8. Global -regularity for 4-dimensional Ricci flow with integral scalar curvature bound

    Wangjian Jian

    Topics 4-manifolds, Scalar curvature

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) 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.