Paper · Apr 11, 2020

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

Yi Lai

math.DGarXiv:2004.05291v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.

View PDF View HTML arXiv page
Submitted
Apr 11, 2020
Last revised
Jun 15, 2020 (v2)
Subjects
math.DG
Journal
Geom. Topol. 25 (2021) 3629-3690
DOI
10.2140/gt.2021.25.3629
Ricci flow3-manifoldsWeak & singular flowsBoundaries & non-compact

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 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)3Kleiner 2014 Singular Ricci flows II (Bruce Kleiner and John Lott, 2018)1Kleiner 2018 Ricci flow under local almost non-negative curvature conditions (Yi Lai, 2018)4Lai 2018 Ricci Solitons, Conical Singularities, and Nonuniqueness (Sigurd B. Angenent and Dan Knopf, 2019)6Angenent 2019 Uniqueness and stability of singular Ricci flows in higher dimensions (Robert Haslhofer, 2021)5Haslhofer 2021 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)7Deruelle 2022 Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows (Albert Chau and Adam Martens, 2023)2Chau 2023 Ends of (singular) Ricci shrinkers (Alessandro Bertellotti and Reto Buzano, 2024)8Bertellotti 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singular Ricci flows II

    Bruce Kleiner and John Lott

    Topics Weak & singular flows

  2. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

    Albert Chau and Adam Martens

    Topics 3-manifolds, Weak & singular flows, Boundaries & non-compact · Shared terms “Kleiner”, “Lott”, “Riemannian manifold”

  3. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon

    Topics 3-manifolds, Weak & singular flows

  4. Ricci flow under local almost non-negative curvature conditions

    Yi Lai

    Topics 3-manifolds · Same author Yi Lai

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

    Robert Haslhofer

    Topics 3-manifolds, Weak & singular flows · Shared terms “Kleiner”

  6. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “Kleiner”, “Lott”

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

    Alix Deruelle, Felix Schulze and Miles Simon

    Topics 3-manifolds, Weak & singular flows · Shared terms “Lott”, “possibly”, “Riemannian manifold”

  8. Ends of (singular) Ricci shrinkers

    Alessandro Bertellotti and Reto Buzano

    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “possibly”

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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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.

  2. 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.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.