Paper · Feb 6, 2020

Wave Approximation of Backward Heat Equation with Ricci Flow

Jie Xu

math.DGarXiv:2002.02350
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Abstract

In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation outside its wavefront set onto solves the backward heat equation within some appropriate time interval. We discuss this approximation starting from Euclidean case, and then extend to the open Riemannian manifold situation. This idea partially comes from Perelman's original papers in proving Poincaré conjecture as well as Terence Tao's Notes in UCLA.

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Submitted
Feb 6, 2020
Subjects
math.DG, math.AP
Ricci flowPoincaré conjecture

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)1Morgan 2006 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)8Cao 2006 The Optimal Double Bubble for Density r^p (Jack Hirsch, Kevin Li, Jackson Petty et al., 2019)3Hirsch 2019 Product structure and regularity theorem for totally nonnegative flag varieties (Huanchen Bao and Xuhua He, 2022)4Bao 2022 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)2Akbulut 2022 Moving Manifolds and the Poincare Conjecture (David V. Svintradze, 2025)7Svintradze 2025 Equations in Products of Free Groups and 3-Manifold Groups, I (Olga Kharlampovich and Alina Vdovina, 2026)5Kharlampovic 2026 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)6Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon

    Topics Poincaré conjecture · Shared terms “Perelman”

  2. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut

    Topics Poincaré conjecture

  3. The Optimal Double Bubble for Density

    Jack Hirsch, Kevin Li, Jackson Petty et al.

    Topics Poincaré conjecture · Shared terms “Euclidean”, “Perelman”

  4. Product structure and regularity theorem for totally nonnegative flag varieties

    Huanchen Bao and Xuhua He

    Topics Poincaré conjecture

  5. Equations in Products of Free Groups and 3-Manifold Groups, I

    Olga Kharlampovich and Alina Vdovina

    Topics Poincaré conjecture · Shared terms “Perelman”

  6. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al.

    Topics Poincaré conjecture · Shared terms “Perelman”

  7. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze

    Topics Poincaré conjecture

  8. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon

    Topics Poincaré conjecture · Shared terms “Perelman”

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)

In the guidebook

  1. 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.

  2. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

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

  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.