Paper · Aug 2, 2017

Ricci Flow of Compact Locally Homogeneous Geometries on 5-Manifolds

Amanda Hirschmann, Thomas Bell

math.DGarXiv:1708.00869
View PDF View HTML arXiv page

As on arXiv

Abstract

This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of Ricci Flow in a certain class of five dimensional homogeneous manifolds.

View PDF View HTML arXiv page
Submitted
Aug 2, 2017
Subjects
math.DG

Comments: Amanda Hirschmann is the primary author

Ricci flowPoincaré conjectureGeometrization4-manifoldsHomogeneous spaces

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 Ricci Flow and the Poincare Conjecture (John W. Morgan and Gang Tian, 2006)4Morgan 2006 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)1Cao 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)5Morgan 2008 Wave Approximation of Backward Heat Equation with Ricci Flow (Jie Xu, 2020)8Xu 2020 Decomposition space theory (Boldizsar Kalmar, 2021)7Kalmar 2021 Classical Poincaré conjecture via 4D topology (Akio Kawauchi, 2021)6Kawauchi 2021 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)2Akbulut 2022 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)3de Freitas 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon

    Topics Poincaré conjecture, Geometrization · Shared terms “Hamilton”

  2. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut

    Topics Poincaré conjecture, 4-manifolds

  3. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization · Shared terms “Richard Hamilton”, “Poincaré conjecture”

  4. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon

    Topics Poincaré conjecture

  5. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon

    Topics Poincaré conjecture, Geometrization

  6. Classical Poincaré conjecture via 4D topology

    Akio Kawauchi

    Topics Geometrization, 4-manifolds · Shared terms “Poincaré conjecture”

  7. Decomposition space theory

    Boldizsar Kalmar

    Topics Poincaré conjecture, 4-manifolds · Shared terms “dimensional manifolds”

  8. Wave Approximation of Backward Heat Equation with Ricci Flow

    Jie Xu

    Topics Poincaré conjecture

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) Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03… Start with: Generalized Poincare's Conjecture in Dimensions Greater Than Four (Smale, 1961) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 10A.10
    Geometrization and Ricci FlowBook 10A · Three-Manifolds and Geometrization

    The fate of each piece under the flow.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

    Every word of the statement, and why each hypothesis is needed.