Paper · Nov 18, 2016

Local control on the geometry in 3D Ricci flow

Miles Simon, Peter M. Topping

math.DGarXiv:1611.06137v2
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Abstract

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local C/t decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman and Tian-Wang in this dimension by not requiring the Ricci curvature to be almost-positive, and not asking the volume growth to be almost-Euclidean.

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Submitted
Nov 18, 2016
Last revised
Jun 30, 2017 (v2)
Subjects
math.DG, math.AP

Comments: Minor updates in order to integrate with arXiv:1706.09490 prior to being submitted

Ricci flow3-manifolds

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 Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (Miles Simon and Peter M. Topping, 2017)1Simon 2017 Global Regularity of Three-Dimensional Ricci Limit Spaces (Andrew D. McLeod and Peter M. Topping, 2018)3McLeod 2018 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)4Deruelle 2022 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)6Lee 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 Volume estimates and convergence results for solutions to Ricci flow with L^(p) bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)7Liu 2024 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)2Deruelle 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)8Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

    Miles Simon and Peter M. Topping

    Topics 3-manifolds · Same authors Miles Simon, Peter M. Topping · Shared terms “Tian”

  2. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al.

    Topics 3-manifolds · Same authors Miles Simon, Peter M. Topping · Shared terms “requiring”

  3. Global Regularity of Three-Dimensional Ricci Limit Spaces

    Andrew D. McLeod and Peter M. Topping

    Topics 3-manifolds · Same author Peter M. Topping · Shared terms “dimensional Ricci”

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

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics 3-manifolds · Same author Miles Simon · Shared terms “dimensional Ricci”, “Riemannian manifold”, “Euclidean”

  5. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics 3-manifolds · Same author Peter M. Topping · Shared terms “generalises”

  6. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping

    Topics 3-manifolds · Same author Peter M. Topping

  7. Volume estimates and convergence results for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon

    Topics Ricci flow · Same author Miles Simon · Shared terms “full curvature”, “dimensional Ricci”, “curvature tensor”

  8. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng

    Topics 3-manifolds · Shared terms “lower bound”

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) 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)

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

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