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Sp(2)-invariant expanders and shrinkers in Laplacian flow

Mark Haskins, Rowan Juneman, Johannes Nordström

TL;DR

This work advances the understanding of cohomogeneity-one Laplacian solitons under Sp$(2)$-action by classifying the complete expanding solitons on $\Lambda^2_-S^4$, showing they form a 1-parameter family parameterized by $q=\lambda\sqrt{\mathrm{Vol}(S^4)}$ and proving all such solitons are asymptotically conical with rate $-2$ to a unique cone. It introduces robust monotonicity tools (notably the warping $h=\frac{y}{x}$ and the adjusted torsion $\tilde{\tau}_1,\tilde{\tau}_2$) to analyze end behaviors, and proves a bijection between the expander parameter $q$ and the asymptotic cone parameter $\ell>1$. The paper also develops a detailed trichotomy for possible ends (including forward-complete AC ends, forward-complete non-AC ends, and forward-incomplete ends) and studies the stability and regularity of AC ends, proving $C^0$ rate $-2$ convergence for non-steady AC ends and continuity of AC cones under perturbations. Additionally, it demonstrates that the torsion-free cone is not an asymptotic cone of any smoothly-closing expander and establishes uniqueness of expanders given their asymptotic cone, while exploring SU$(3)$-invariant solitons and conjectures about conifold-type transitions linking AC expanders and shrinkers in that setting. The results illuminate how Laplacian flow may implement a form of surgery at conical singularities and provide a framework for comparing AC solitons across symmetry classes with potential implications for flow continuation and singularity models.

Abstract

We show that the complete Sp(2)-invariant expanding solitons for Bryant's Laplacian flow on the anti-self-dual bundle of the 4-sphere form a 1-parameter family, and that they are all asymptotically conical (AC). We determine their asymptotic cones, and prove that this cone determines the complete expander (up to scale). Neither the unique Sp(2)-invariant torsion-free G_2-cone nor the asymptotic cone of the explicit AC Sp(2)-invariant shrinker from arxiv:2112.09095 occurs as the asymptotic cone of a complete AC Sp(2)-invariant expander. We determine all possible end behaviours of Sp(2)-invariant solitons, identifying novel forward-complete end solutions for both expanders and shrinkers with faster-than-Euclidean volume growth. We conjecture that there exists a 1-parameter family of complete SU(3)-invariant expanders on the anti-self-dual bundle of the complex projective plane CP^2 with such asymptotic behaviour. We also conjecture that, in contrast to the Sp(2)-invariant case, there exist complete SU(3)-invariant AC expanders with asymptotic cone matching that of the explicit AC SU(3)-invariant shrinker from arxiv:2112.09095. The latter conjecture suggests that Laplacian flow may naturally implement a type of surgery in which a CP^2 shrinks to a conically singular point, but after which the flow can be continued smoothly, expanding a topologically different CP^2 from the singularity.

Sp(2)-invariant expanders and shrinkers in Laplacian flow

TL;DR

This work advances the understanding of cohomogeneity-one Laplacian solitons under Sp-action by classifying the complete expanding solitons on , showing they form a 1-parameter family parameterized by and proving all such solitons are asymptotically conical with rate to a unique cone. It introduces robust monotonicity tools (notably the warping and the adjusted torsion ) to analyze end behaviors, and proves a bijection between the expander parameter and the asymptotic cone parameter . The paper also develops a detailed trichotomy for possible ends (including forward-complete AC ends, forward-complete non-AC ends, and forward-incomplete ends) and studies the stability and regularity of AC ends, proving rate convergence for non-steady AC ends and continuity of AC cones under perturbations. Additionally, it demonstrates that the torsion-free cone is not an asymptotic cone of any smoothly-closing expander and establishes uniqueness of expanders given their asymptotic cone, while exploring SU-invariant solitons and conjectures about conifold-type transitions linking AC expanders and shrinkers in that setting. The results illuminate how Laplacian flow may implement a form of surgery at conical singularities and provide a framework for comparing AC solitons across symmetry classes with potential implications for flow continuation and singularity models.

Abstract

We show that the complete Sp(2)-invariant expanding solitons for Bryant's Laplacian flow on the anti-self-dual bundle of the 4-sphere form a 1-parameter family, and that they are all asymptotically conical (AC). We determine their asymptotic cones, and prove that this cone determines the complete expander (up to scale). Neither the unique Sp(2)-invariant torsion-free G_2-cone nor the asymptotic cone of the explicit AC Sp(2)-invariant shrinker from arxiv:2112.09095 occurs as the asymptotic cone of a complete AC Sp(2)-invariant expander. We determine all possible end behaviours of Sp(2)-invariant solitons, identifying novel forward-complete end solutions for both expanders and shrinkers with faster-than-Euclidean volume growth. We conjecture that there exists a 1-parameter family of complete SU(3)-invariant expanders on the anti-self-dual bundle of the complex projective plane CP^2 with such asymptotic behaviour. We also conjecture that, in contrast to the Sp(2)-invariant case, there exist complete SU(3)-invariant AC expanders with asymptotic cone matching that of the explicit AC SU(3)-invariant shrinker from arxiv:2112.09095. The latter conjecture suggests that Laplacian flow may naturally implement a type of surgery in which a CP^2 shrinks to a conically singular point, but after which the flow can be continued smoothly, expanding a topologically different CP^2 from the singularity.
Paper Structure (44 sections, 70 theorems, 218 equations, 2 figures)

This paper contains 44 sections, 70 theorems, 218 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Laplacian flow and its solitons
  3. Complete expanders and their asymptotic cones
  4. Matching AC expanders with shrinkers and G_2 conifold transitions
  5. Soliton ends
  6. Structure of the paper
  7. SU(3)-invariant expanders and shrinkers
  8. Further questions
  9. The G_2-soliton equations
  10. Singularity formation and resolution in Laplacian flow and Laplacian solitons
  11. Sp(2)-invariant Laplacian solitons
  12. First-order reformulation of the soliton ODE
  13. Adjusted torsion
  14. Complete expanders
  15. Solitons extending smoothly over the singular orbit
  16. ...and 29 more sections

Key Result

Theorem 1

There exists (up to scaling) a $1$-parameter family of complete $\textup{Sp$(2)$}$-invariant gradient Laplacian expanders on $\Lambda^2_-\mathbb{S}^4$. The scale-invariant parameter is $q = \lambda \sqrt{\operatorname{Vol}(\mathbb{S}^4)} \in \mathbb{R}_{>0}$, where $\operatorname{Vol}(\mathbb{S}^4)$

Figures (2)

  • Figure 1: Phase diagram of the shrinker limit system \ref{['eq:heisenberg:limit']}
  • Figure 2: Closed SU$(3)$-invariant cones

Theorems & Definitions (179)

  • Remark 1.1
  • Theorem 1
  • Remark 1.4
  • Definition 1.8
  • Theorem 2
  • Example 1.9
  • Corollary 3
  • Conjecture 1
  • Theorem 4: Bounded warping implies AC with rate $-2$
  • Theorem 5
  • ...and 169 more