Paper · Jun 3, 2021

Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

Gilles Carron, Jørgen Olsen Lye, Boris Vertman

math.DGWider flowsarXiv:2106.01799
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Abstract

In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low energy condition. We also prove a concentration - compactness dichotomy, and investigate what the alternatives to convergence are. We end by investigating a non-convergent example due to Viaclovsky in more detail.

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Submitted
Jun 3, 2021
Subjects
math.DG, math.AP
Journal
Tohoku Math. J. (2) 75(4): 561-615 (2023)
DOI
10.2748/tmj.20220616

Comments: 54 pages, 1 figure

Yamabe flowWeak & singular flowsLimit spaces & RCD

Its neighbourhood

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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 Long-time existence of the edge Yamabe flow (Eric Bahuaud and Boris Vertman, 2016)4Bahuaud 2016 Convergence of the CR Yamabe Flow (Pak Tung Ho, Weimin Sheng and Kunbo Wang, 2017)7Ho 2017 Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant (Jørgen Olsen Lye and Boris Vertman, 2020)1Lye 2020 CR Yamabe constant, CR Yamabe flow and its soliton (Pak Tung Ho and Kunbo Wang, 2020)6Ho 2020 Normalized Yamabe flow on manifolds with bounded geometry (Bruno Caldeira, Luiz Hartmann and Boris Vertman, 2021)5Caldeira 2021 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)2Carron 2022 Convergence rate of the weighted Yamabe flow (Pak Tung Ho, Jinwoo Shin and Zetian Yan, 2022)8Ho 2022 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)3Lye 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-time existence of Yamabe flow on singular spaces with positive Yamabe constant

    Jørgen Olsen Lye and Boris VertmanWider flows

    Topics Yamabe flow, Weak & singular flows · Same authors Jørgen Olsen Lye, Boris Vertman · Shared terms “positive Yamabe”, “Yamabe constant”, “singular spaces”

  2. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Yamabe flow · Same author Gilles Carron · Shared terms “Yamabe constant”, “positive Yamabe”

  3. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow · Same authors Jørgen Olsen Lye, Boris Vertman

  4. Long-time existence of the edge Yamabe flow

    Eric Bahuaud and Boris VertmanWider flows

    Topics Yamabe flow · Same author Boris Vertman · Shared terms “edge metrics”, “normalized Yamabe”

  5. Normalized Yamabe flow on manifolds with bounded geometry

    Bruno Caldeira, Luiz Hartmann and Boris VertmanWider flows

    Topics Yamabe flow · Same author Boris Vertman · Shared terms “normalized Yamabe”

  6. CR Yamabe constant, CR Yamabe flow and its soliton

    Pak Tung Ho and Kunbo WangWider flows

    Topics Yamabe flow · Shared terms “Yamabe constant”

  7. Convergence of the CR Yamabe Flow

    Pak Tung Ho, Weimin Sheng and Kunbo WangWider flows

    Topics Yamabe flow

  8. Convergence rate of the weighted Yamabe flow

    Pak Tung Ho, Jinwoo Shin and Zetian YanWider flows

    Topics Yamabe flow, Limit spaces & RCD

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.

Foundational paper in the Yamabe flow guide: Carron–Lye–Vertman: singular spaces with positive Yamabe constant, and what the closed-manifold theorems cover.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.