Paper · Oct 14, 2022

Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow

Mark Haskins, Ilyas Khan, Alec Payne

math.DGarXiv:2210.07954
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove a uniqueness result for asymptotically conical (AC) gradient shrinking solitons for the Laplacian flow of closed G_2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G_2-cone, then their G_2-structures are equivalent, and in particular, the two solitons are isometric. The proof extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons. We additionally show that the symmetries of the G_2-structure of an AC shrinker end are inherited from its asymptotic cone; under a mild assumption on the fundamental group, the symmetries of the asymptotic cone extend to global symmetries.

View PDF View HTML arXiv page
Submitted
Oct 14, 2022
Subjects
math.DG, math.AP

Comments: 80 pages, Comments welcome

Generalized & G₂ flowsRicci solitonsStability & uniquenessBoundaries & 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 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 Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers (Maxwell Stolarski, 2022)5Stolarski 2022 Backward propagation of warped product structures and asymptotically conical shrinkers (Brett Kotschwar, 2023)2Kotschwar 2023 On the shrinking solitons of generalized Ricci flow (Xilun Li and Yanan Ye, 2024)1Li 2024 Dynamical stability of Pluriclosed and Generalized Ricci solitons (Kuan-Hui Lee, 2025)7Lee 2025 On steady and expanding Ricci solitons with asymptotic symmetries (Michael B. Law, 2025)4Law 2025 Stability of asymptotically conical gradient Kähler-Ricci expanders (Longteng Chen, 2025)3Chen 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)8Conlon 2026 Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow (Shaochuang Huang and Zhuo Peng, 2026)6Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the shrinking solitons of generalized Ricci flow

    Xilun Li and Yanan Ye

    Topics Generalized & G₂ flows, Ricci solitons · Shared terms “shrinking solitons”, “gradient shrinking”

  2. Backward propagation of warped product structures and asymptotically conical shrinkers

    Brett Kotschwar

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “asymptotically conical”, “conical gradient”, “shrinking solitons”

  3. Stability of asymptotically conical gradient Kähler-Ricci expanders

    Longteng Chen

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “conical gradient”, “asymptotically conical”

  4. On steady and expanding Ricci solitons with asymptotic symmetries

    Michael B. Law

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “global symmetries”, “asymptotically conical”

  5. Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

    Maxwell Stolarski

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “asymptotically conical”

  6. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng

    Topics Generalized & G₂ flows, Boundaries & non-compact, Stability & uniqueness

  7. Dynamical stability of Pluriclosed and Generalized Ricci solitons

    Kuan-Hui Lee

    Topics Generalized & G₂ flows, Stability & uniqueness, Ricci solitons

  8. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “asymptotically conical”

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.

Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.