Paper · Apr 25, 2024

Willmore-type inequalities for closed hypersurfaces in weighted manifolds

Guoqiang Wu, Jia-Yong Wu

math.DGarXiv:2404.16286v5
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Abstract

In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature. In particular, we give a sharp Willmore type inequality in steady gradient Ricci solitons. We also prove a sharp Willmore-like inequality in shrinking gradient Ricci solitons. Moreover, we characterize the equality cases of Willmore-type inequalities. These results can be regarded as weighted versions of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality. As applications, we derive some sharp isoperimetric type inequalities in weighted manifolds under the existence assumption of a critical set of weighted isoperimetric functional.

Comments: Final version, accepted by J. Geom. Anal

Ricci solitonsLimit spaces & RCDGeometric inequalities

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of weighted manifolds via classification results for semilinear equations

    Giulio Ciraolo, Alberto Farina and Troy Petitt PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “weighted manifolds”, “Émery Ricci”, “Bakry Émery”

  2. A Comprehensive Review of Solitonic Inequalities in Riemannian Geometry

    Bang-Yen Chen, Majid Ali Choudhary, Mohammed Nisar et al. PDF

    Topics Geometric inequalities, Ricci solitons

  3. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “Émery Ricci”

  4. Sharp Minkowski-Type Inequality in Cartan-Hadamard 3-Spaces

    Fang HongWider flows PDF

    Topics Geometric inequalities · Shared terms “type inequality”

  5. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “Émery Ricci”, “Bakry Émery”, “gradient Ricci”

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

    Liang Cheng PDF

    Topics Geometric inequalities · Shared terms “weighted isoperimetric”

  7. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “shrinking gradient”, “gradient Ricci”

  8. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Ricci solitons · Same author Guoqiang Wu · Shared terms “shrinking gradient”, “gradient Ricci”

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.