Paper · Apr 2, 2020

A steady length function for Ricci flow

Joshua Jordan

math.DGarXiv:2004.00785
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Abstract

A fundamental step in the analysis of singularities of Ricci flow was the discovery by Perelman of a monotonic volume quantity which detected shrinking solitons in (arXiv:math/0211159). A similar quantity was found by Feldman, Ilmanen, and Ni in 2005 which detected expanding solitons. The current work introduces a modified length functional as a first step towards a steady soliton monotonicity formula. This length functional generates a distance function in the usual way which is shown to satisfy several differential inequalities which saturate precisely on manifolds satisfying a modification of the steady soliton equation.

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Submitted
Apr 2, 2020
Subjects
math.DG

Comments: 9 pages

Ricci flowRicci solitonsSingularities

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 Yamabe Flow: steady solitons and type II singularities (Beomjun Choi and Panagiota Daskalopoulos, 2017)1Choi 2017 Ricci Solitons, Conical Singularities, and Nonuniqueness (Sigurd B. Angenent and Dan Knopf, 2019)7Angenent 2019 Four-Dimensional Steady Gradient Ricci Solitons with 3-Cylindrical Tangent Flows at Infinity (Richard Bamler, Bennett Chow, Yuxing Deng et al., 2021)4Bamler 2021 On the H¹(ds)-gradient flow for the length functional (Philip Schrader, Glen Wheeler and Valentina-Mira Wheeler, 2021)2Schrader 2021 Lower bounds for the scalar curvatures of Ricci flow singularity models (Pak-Yeung Chan, Bennett Chow, Zilu Ma et al., 2022)8Chan 2022 Dimension Reduction for Positively Curved Steady Solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)6Chan 2023 Unique Asymptotics of Steady Ricci Solitons with Symmetry (Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum, 2023)3Ma 2023 Homogeneous Sobolev gradient flow of the length functional (Philip Schrader, Glen Wheeler and Valentina Wheeler, 2026)5Schrader 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Yamabe Flow: steady solitons and type II singularities

    Beomjun Choi and Panagiota Daskalopoulos

    Topics Singularities, Ricci solitons · Shared terms “steady soliton”

  2. On the -gradient flow for the length functional

    Philip Schrader, Glen Wheeler and Valentina-Mira WheelerWider flows

    Shared terms “length functional”

  3. Unique Asymptotics of Steady Ricci Solitons with Symmetry

    Zilu Ma, Hamidreza Mahmoudian and Natasa Sesum

    Topics Ricci solitons · Shared terms “step towards”

  4. Four-Dimensional Steady Gradient Ricci Solitons with -Cylindrical Tangent Flows at Infinity

    Richard Bamler, Bennett Chow, Yuxing Deng et al.

    Topics Singularities, Ricci solitons · Shared terms “steady soliton”

  5. Homogeneous Sobolev gradient flow of the length functional

    Philip Schrader, Glen Wheeler and Valentina WheelerWider flows

    Topics Singularities · Shared terms “length functional”

  6. Dimension Reduction for Positively Curved Steady Solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Singularities, Ricci solitons · Shared terms “steady soliton”

  7. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    Topics Singularities, Ricci solitons · Shared terms “expanding solitons”, “shrinking soliton”

  8. Lower bounds for the scalar curvatures of Ricci flow singularity models

    Pak-Yeung Chan, Bennett Chow, Zilu Ma et al.

    Topics Singularities, Ricci solitons · Shared terms “steady soliton”, “monotonicity formula”

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) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.