Paper · Jan 12, 2025

Rigidity Results Involving Stabilized Scalar Curvature

Yipeng Wang

math.DGarXiv:2501.06951
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Abstract

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of -stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the -stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.

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Submitted
Jan 12, 2025
Subjects
math.DG

Comments: 26 pages

Ricci flowScalar curvatureStability & uniquenessFree boundary & capillaryMinimal & CMC surfaces

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 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 Free boundary flow with surgery (Robert Haslhofer, 2023)7Haslhofer 2023 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)1Haslhofer 2023 On local rigidity theorems with respect to the scalar curvature (Liang Cheng, 2023)8Cheng 2023 Scalar curvature rigidity of the four-dimensional sphere (Simone Cecchini, Jinmin Wang, Zhizhang Xie et al., 2024)5Cecchini 2024 Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds (Liam Mazurowski and Jintian Zhu, 2025)2Mazurowski 2025 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)4Sun 2025 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)6Kroencke 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)3Bourni 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “foliation”

  2. Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

    Liam Mazurowski and Jintian ZhuWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Scalar curvature · Shared terms “constant mean”

  3. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary, Stability & uniqueness · Shared terms “foliation”

  4. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu

    Topics Minimal & CMC surfaces, Scalar curvature

  5. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al.

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

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

    Klaus Kroencke

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

  7. Free boundary flow with surgery

    Robert HaslhoferWider flows

    Topics Minimal & CMC surfaces, Free boundary & capillary

  8. On local rigidity theorems with respect to the scalar curvature

    Liang Cheng

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

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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

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.