Paper · Nov 10, 2019

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

Shun Maeta

math.DGarXiv:1911.03902v6
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Abstract

Let be a 3-dimensional complete steady gradient Ricci soliton. Assume that is rectifiable, that is, the potential function can be written as , where is a distance function. Then, we prove that is isometric to (1) a quotient of , or (2) the Bryant soliton. In particular, we show that any 3-dimensional complete rectifiable steady gradient Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. Furthermore, we show that any -dimensional complete rectifiable expanding gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.

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Submitted
Nov 10, 2019
Last revised
Sep 17, 2023 (v6)
Subjects
math.DG

Comments: The author found a mistake

Ricci solitonsAncient solutions3-manifoldsRotational symmetry & warped products

Its neighbourhood

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

    Yi Lai

    Topics Ancient solutions, 3-manifolds, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “Bryant soliton”

  2. 3d steady Gradient Ricci Solitons with linear curvature decay

    Yuxing Deng and Xiaohua Zhu

    Topics Rotational symmetry & warped products, 3-manifolds, Ricci solitons · Shared terms “steady gradient”, “gradient Ricci”, “rotationally symmetric”

  3. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

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

  4. On the construction of complete expanding gradient Rici solitons

    Marcio de Sousa, Paula Bonfim, Romildo Pina et al.

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “expanding gradient”, “gradient Ricci”, “potential function”

  5. Hamilton-Ivey estimates for gradient Ricci solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Rotational symmetry & warped products, 3-manifolds, Ricci solitons · Shared terms “gradient Ricci”, “expanding gradient”, “steady gradient”

  6. A new proof for the existence of rotationally symmetric gradient Ricci solitons

    Shu-Yu Hsu

    Topics Rotational symmetry & warped products, Ricci solitons · Shared terms “expanding gradient”, “gradient Ricci”, “rotationally symmetric”

  7. The three-dimensional standard solution to the Ricci flow is modeled by the Bryant soliton

    Yongjia Zhang

    Topics Ancient solutions, 3-manifolds · Shared terms “Bryant soliton”

  8. Gradient Ricci solitons with vanishing conditions on Weyl

    Giovanni Catino, Paolo Mastrolia and Dario Daniele Monticelli

    Topics Ancient solutions, 3-manifolds, Ricci solitons · Shared terms “gradient Ricci”, “Bryant soliton”, “quotient”

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) 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) 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)

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.