Paper · Nov 21, 2022

Complete -dimensional Ricci flow spacetimes

Luke Thomas Peachey

math.DGarXiv:2211.11866
PDF Abstract page

As on arXiv

Abstract

Ricci flow spacetimes were introduced by Kleiner & Lott as a way to describe Ricci flow through singularities, and have since been used elsewhere in the literature, prompting the question of their rigidity. In -dimensions, we show that every complete and sufficiently regular spacetime must be a cylindrical spacetime. That is, if the metric is complete on each spatial slice, after imposing a necessary continuity condition, we can conclude that every spatial slice must be diffeomorphic to a fixed surface, and the Ricci flow spacetime is isometric to a classical Ricci flow on this surface.

Ricci flowSingularitiesWeak & singular flowsStability & uniqueness

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 2021, 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 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)6Kleiner 2014 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)7Haslhofer 2021 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)2Deruelle 2022 All two-dimensional expanding Ricci solitons (Luke T. Peachey and Peter M. Topping, 2023)4Peachey 2023 The weighted ambient metric for manifolds with density (Ayush Khaitan, 2023)1Khaitan 2023 Almost splitting and quantitative stratification for super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2023)5Kunikawa 2023 On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces (Richard H Bamler and Bruce Kleiner, 2023)8Bamler 2023 Expanding Ricci solitons coming out of weakly PIC1 metric cones (Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey, 2024)3Chan 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The weighted ambient metric for manifolds with density

    Ayush Khaitan PDF

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “flow spacetime”, “Kleiner Lott”

  2. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “dimensional Ricci”, “Lott”

  3. Expanding Ricci solitons coming out of weakly PIC1 metric cones

    Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey PDF

    Topics Weak & singular flows · Same author Luke Thomas Peachey

  4. All two-dimensional expanding Ricci solitons

    Luke T. Peachey and Peter M. Topping PDF

    Topics Stability & uniqueness · Same author Luke Thomas Peachey

  5. Almost splitting and quantitative stratification for super Ricci flow

    Keita Kunikawa and Yohei Sakurai PDF

    Topics Weak & singular flows, Stability & uniqueness · Shared terms “conclude”

  6. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Weak & singular flows · Shared terms “flow spacetime”

  7. Uniqueness and stability of singular Ricci flows in higher dimensions

    Robert Haslhofer PDF

    Topics Weak & singular flows, Singularities, Stability & uniqueness · Shared terms “dimensional Ricci”, “Kleiner”

  8. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Weak & singular flows, Singularities · Shared terms “diffeomorphic”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.