Paper · Apr 22, 2018

Ricci flow under local almost non-negative curvature conditions

Yi Lai

math.DGarXiv:1804.08073v3
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Abstract

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly cases, of which the results are new. We complete the discussion of the almost preservation problem by Bamler-Cabezas-Rivas-Wilking, and the 2-non-negative case generalizes a result in 3D by Simon-Topping to higher dimensions. As an application, we use the local Ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non-negative curvature conditions, and show that this limit space is bi-Hlder homeomorphic to a smooth manifold.

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Submitted
Apr 22, 2018
Last revised
Jun 12, 2018 (v3)
Subjects
math.DG

Comments: 45 pages, 1 figure

Ricci flow3-manifoldsSphere theorems

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 The Ricci flow under almost non-negative curvature conditions (Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking, 2017)8Bamler 2017 Pyramid Ricci Flow in Higher Dimensions (Andrew D. McLeod and Peter M. Topping, 2019)5McLeod 2019 Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows (Yi Lai, 2020)2Lai 2020 A family of 3d steady gradient solitons that are flying wings (Yi Lai, 2020)1Lai 2020 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)3Lee 2022 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)7Bourni 2023 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)4Abiev 2023 A family of Kähler flying wing steady Ricci solitons (Pak-Yeung Chan, Ronan J. Conlon and Yi Lai, 2024)6Chan 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A family of 3d steady gradient solitons that are flying wings

    Yi Lai

    Topics Sphere theorems, 3-manifolds · Same author Yi Lai

  2. Producing 3d Ricci flows with non-negative Ricci curvature via singular Ricci flows

    Yi Lai

    Topics 3-manifolds · Same author Yi Lai

  3. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping

    Topics Sphere theorems, 3-manifolds · Shared terms “Topping”, “Simon”

  4. On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces

    Nurlan Abiev

    Topics Sphere theorems, 3-manifolds

  5. Pyramid Ricci Flow in Higher Dimensions

    Andrew D. McLeod and Peter M. Topping

    Topics Sphere theorems · Shared terms “limit space”, “higher dimensions”, “smooth manifold”

  6. A family of Kähler flying wing steady Ricci solitons

    Pak-Yeung Chan, Ronan J. Conlon and Yi Lai

    Topics Sphere theorems · Same author Yi Lai

  7. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al.

    Topics Sphere theorems, 3-manifolds

  8. The Ricci flow under almost non-negative curvature conditions

    Richard H. Bamler, Esther Cabezas-Rivas and Burkhard Wilking

    Topics Ricci flow · Shared terms “negative curvature”, “curvature condition”, “almost preservation”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 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.

  3. 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.