Paper · Sep 1, 2017

Level set flow in 3D steady gradient Ricci solitons

Chih-Wei Chen, Kuo-Wei Lee

math.DGarXiv:1709.00143
View PDF View HTML arXiv page

As on arXiv

Abstract

Let be a nontrivial 3-dimensional steady gradient Ricci soliton. If the scalar curvature satisfies for some , and , then the umbilical ratio of the level sets of satisfies .

View PDF View HTML arXiv page
Submitted
Sep 1, 2017
Subjects
math.DG
Ricci solitons3-manifoldsScalar curvatureWeak & singular flows

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 3d steady Gradient Ricci Solitons with linear curvature decay (Yuxing Deng and Xiaohua Zhu, 2016)1Deng 2016 Regularity and stability results for the level set flow via the mean curvature flow with surgery (Alexander Mramor, 2017)6Mramor 2017 Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons (Shun Maeta, 2019)8Maeta 2019 A criterion for the triviality of non-steady gradient Ricci solitons (Mohammed Guediri, 2020)2Guediri 2020 A family of 3d steady gradient solitons that are flying wings (Yi Lai, 2020)7Lai 2020 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)5Sun 2022 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)3Lai 2022 Notes on scalar curvature lower bounds of steady gradient Ricci solitons (Shota Hamanaka, 2024)4Hamanaka 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 3d steady Gradient Ricci Solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics 3-manifolds, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “dimensional steady”

  2. A criterion for the triviality of non-steady gradient Ricci solitons

    Mohammed Guediri

    Topics Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”

  3. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai

    Topics 3-manifolds, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”

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

    Shota Hamanaka

    Topics Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”

  5. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue

    Topics Weak & singular flows, Ricci solitons · Shared terms “set flow”, “level set”

  6. Regularity and stability results for the level set flow via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Weak & singular flows · Shared terms “set flow”, “level set”

  7. A family of 3d steady gradient solitons that are flying wings

    Yi Lai

    Topics 3-manifolds, Scalar curvature, Ricci solitons · Shared terms “steady gradient”, “dimensional steady”, “gradient Ricci”

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

    Shun Maeta

    Topics 3-manifolds, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “Let”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.