Paper · Dec 11, 2019

The isoperimetric inequality in steady Ricci solitons

Yuqiao Li

math.DGarXiv:1912.05250
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.

View PDF View HTML arXiv page
Submitted
Dec 11, 2019
Subjects
math.DG

Comments: 6 pages

Ricci solitonsAncient solutionsRotational symmetry & warped productsGeometric inequalities

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 Steady Ricci solitons with horizontally ε-pinched Ricci curvature (Yuxing Deng and Xiaohua Zhu, 2016)8Deng 2016 Isoperimetric inequalities for eigenvalues by inverse mean curvature flow (Fangcheng Guo, Guanghan Li and Chuanxi Wu, 2016)4Guo 2016 A Curvature Flow and Applications to an Isoperimetric Inequality (Dylan Cant, 2016)1Cant 2016 Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds (Fabio Cavalletti and Andrea Mondino, 2017)6Cavalletti 2017 Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons (Shun Maeta, 2019)5Maeta 2019 On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor (Huai-Dong Cao and Jiangtao Yu, 2020)7Cao 2020 Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (Fengjiang Li, 2021)2Li 2021 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)3Leandro 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Curvature Flow and Applications to an Isoperimetric Inequality

    Dylan CantWider flows

    Topics Geometric inequalities, Rotational symmetry & warped products · Shared terms “isoperimetric inequality”

  2. Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature

    Fengjiang Li

    Topics Rotational symmetry & warped products, Ancient solutions, Ricci solitons · Shared terms “steady Ricci”, “Bryant”

  3. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Ancient solutions, Ricci solitons · Shared terms “steady Ricci”, “steady soliton”, “Bryant”

  4. Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

    Fangcheng Guo, Guanghan Li and Chuanxi WuWider flows

    Topics Geometric inequalities · Shared terms “isoperimetric inequality”

  5. Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons

    Shun Maeta

    Topics Rotational symmetry & warped products, Ancient solutions, Ricci solitons · Shared terms “Bryant”

  6. Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds

    Fabio Cavalletti and Andrea Mondino

    Topics Geometric inequalities · Shared terms “isoperimetric inequality”, “Riemannian manifold”

  7. On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

    Huai-Dong Cao and Jiangtao Yu

    Topics Ancient solutions, Ricci solitons · Shared terms “steady Ricci”, “Bryant”

  8. Steady Ricci solitons with horizontally -pinched Ricci curvature

    Yuxing Deng and Xiaohua Zhu

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “steady 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994) 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. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.