Paper · Jun 14, 2022

Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers

Reto Buzano, Louis Yudowitz

math.DGarXiv:2206.06791v2Math. Z. 304, 7 (2023)
PDF Abstract page

As on arXiv

Abstract

We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer-Mueller. In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

Comments: 32 pages; v2: Final version after minor revision, to appear in Math. Z

Ricci solitonsLimit spaces & RCDDiscrete & graph Ricci

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 2021, 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 A version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)5Hua 2024 Ends of (singular) Ricci shrinkers (Alessandro Bertellotti and Reto Buzano, 2024)4Bertellotti 2024 Semi-Continuity of the Morse Index for Ricci Shrinkers (Louis Yudowitz, 2024)1Yudowitz 2024 Geometric Structure of Ends of Ricci Shrinkers (Alessandro Bertellotti and Reto Buzano, 2025)6Bertellotti 2025 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)2Bai 2025 The weighted Forman and Lin-Lu-Yau Ricci flow on graphs (Shuliang Bai, Shuang Liu and Xin Lai, 2026)8Bai 2026 The character of ideal circle patterns (Chang Li, Aijin Lin and Liangming Shen, 2026)7Li 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)3Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Semi-Continuity of the Morse Index for Ricci Shrinkers

    Louis Yudowitz PDF

    Topics Discrete & graph Ricci, Ricci solitons · Same author Louis Yudowitz · Shared terms “bubble tree”, “Ricci shrinkers”, “sequence”

  2. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al. PDF

    Topics Discrete & graph Ricci

  3. Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci

  4. Ends of (singular) Ricci shrinkers

    Alessandro Bertellotti and Reto Buzano PDF

    Topics Ricci solitons · Same author Reto Buzano · Shared terms “Ricci shrinkers”, “sequence”

  5. A version of Bakry-Émery Ricci flow on a finite graph

    Bobo Hua, Yong Lin and Tao Wang PDF

    Topics Discrete & graph Ricci, Limit spaces & RCD

  6. Geometric Structure of Ends of Ricci Shrinkers

    Alessandro Bertellotti and Reto Buzano PDF

    Topics Ricci solitons · Same author Reto Buzano · Shared terms “Ricci shrinkers”, “sequence”

  7. The character of ideal circle patterns

    Chang Li, Aijin Lin and Liangming Shen PDF

    Topics Discrete & graph Ricci, Limit spaces & RCD · Shared terms “compactness”

  8. The weighted Forman and Lin-Lu-Yau Ricci flow on graphs

    Shuliang Bai, Shuang Liu and Xin Lai PDF

    Topics Discrete & graph Ricci

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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.