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Cohomogeneity one 4-dimensional gradient Ricci solitons

Patrick Donovan

TL;DR

The paper develops a systematic framework for simply-connected 4-manifolds with gradient Ricci solitons under cohomogeneity-one SU(2) symmetry. By reducing the soliton equation to an ODE system on the regular orbits and imposing precise smoothness boundary conditions at singular orbits, it yields new expanding soliton families on $\mathbb{R}^4$ and $\mathcal{O}(-n)$, including degenerate and $\text{SO}(4)$-invariant cases, and classifies complete Einstein expanders within these symmetric classes. It also advances shrinking solitons by constructing and classifying $U(2)$-invariant shrinking Kähler solitons with orbifold singularities, culminating in a Kähler classification (Theorem $\text{thm_Kahlerclassification}$) that encompasses orbifold spaces such as $M_n$, $\mathcal{O}(-n)$, and $\mathbb{CP}^2$. Complementary numerical investigations provide evidence that all compact cohomogeneity-one shrinking solitons are already known within this symmetry class, and they illustrate the dense occurrence of Einstein metrics on the boundary of the expansion parameter spaces. Overall, the work expands the catalog of explicit gradient Ricci solitons in dimension four with SU(2) symmetry and clarifies how symmetry constrains the soliton landscape, with implications for Ricci flow singularity models and orbifold geometries.

Abstract

Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is $SU(2)$ (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected $SU(2)$-invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete $SU(2)$-invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler $U(2)$-invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known.

Cohomogeneity one 4-dimensional gradient Ricci solitons

TL;DR

The paper develops a systematic framework for simply-connected 4-manifolds with gradient Ricci solitons under cohomogeneity-one SU(2) symmetry. By reducing the soliton equation to an ODE system on the regular orbits and imposing precise smoothness boundary conditions at singular orbits, it yields new expanding soliton families on and , including degenerate and -invariant cases, and classifies complete Einstein expanders within these symmetric classes. It also advances shrinking solitons by constructing and classifying -invariant shrinking Kähler solitons with orbifold singularities, culminating in a Kähler classification (Theorem ) that encompasses orbifold spaces such as , , and . Complementary numerical investigations provide evidence that all compact cohomogeneity-one shrinking solitons are already known within this symmetry class, and they illustrate the dense occurrence of Einstein metrics on the boundary of the expansion parameter spaces. Overall, the work expands the catalog of explicit gradient Ricci solitons in dimension four with SU(2) symmetry and clarifies how symmetry constrains the soliton landscape, with implications for Ricci flow singularity models and orbifold geometries.

Abstract

Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected -invariant expanding and shrinking cohomogeneity one gradient Ricci solitons. The first result is the construction of new 3-parameter families of complete -invariant asymptotically conical expanding gradient Ricci solitons. New shrinking Kähler -invariant gradient Ricci solitons in dimension 4 with orbifold singularities are also constructed, leading to a classification of such metrics when the base space of the orbifold is a simply-connected smooth manifold. Finally, we highlight numerical evidence that all the compact cohomogeneity one shrinking gradient Ricci solitons are known.

Paper Structure

This paper contains 15 sections, 29 theorems, 108 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Cohomogeneity one group actions
  3. The initial value problem for $\mathrm{SU}(2)$-invariant gradient Ricci solitons in dimension 4
  4. Constructing the initial value problem
  5. Short-time existence
  6. $\mathrm{SU}(2)$-invariant expanding solitons
  7. Degenerate $\mathrm{U}(2)$-invariant solutions on $\mathcal{O}(-n)$
  8. $\mathrm{SO}(4)$-invariant solitons on $\mathbb{R}^4$
  9. Einstein metrics
  10. Expanding non-Einstein solitons
  11. $\mathrm{SU}(2)$-invariant compact solitons
  12. Examples
  13. Numerical procedure and results
  14. $\mathrm{U}(2)$-invariant shrinking gradient Kähler Ricci solitons with orbifold singularities
  15. Appendix: additional figures for compact shrinkers

Key Result

Theorem 1.1

There exists a $2$-parameter family of complete $\mathrm{U}(2)$-invariant expanding gradient Ricci solitons containing a $1$-parameter family of Einstein metrics on $\mathcal{O}(-n)$ for all $n\in \mathbb{Z}_{>0}$. Moreover, there exists a $3$-parameter family of complete $\mathrm{SU}(2)$-invariant

Figures (6)

  • Figure 1: Numerical simulation of $\Omega_\cdot^E$
  • Figure 2: Numerical simulation of $\Omega$
  • Figure 3: Numerical simulation of $\Omega_n$
  • Figure 4: Numerical simulation of compact solitons
  • Figure 5: Closing conditions for compact Kähler solitons with one smooth end
  • ...and 1 more figures

Theorems & Definitions (80)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • ...and 70 more