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Cohomogeneity one Ricci Solitons from Hopf Fibrations

Matthias Wink

Abstract

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $Ad_K$-invariant irreducible summands, the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano-Dancer-Gallaugher-Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to Böhm. The techniques also apply to $m$-quasi-Einstein metrics.

Cohomogeneity one Ricci Solitons from Hopf Fibrations

Abstract

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit consists of two inequivalent -invariant irreducible summands, the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano-Dancer-Gallaugher-Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to Böhm. The techniques also apply to -quasi-Einstein metrics.

Paper Structure

This paper contains 24 sections, 39 theorems, 115 equations, 1 table.

Table of Contents

  1. The cohomogeneity one Ricci soliton equation
  2. The general set-up
  3. Ricci solitons with a singular orbit
  4. Consequences of completeness
  5. The Böhm functional
  6. New Examples of Ricci solitons
  7. The geometric set-up
  8. Qualitative ODE analysis
  9. Ricci solitons from circle bundles
  10. Asymptotics
  11. Cone solutions
  12. Steady Ricci solitons
  13. Expanding Ricci solitons
  14. Ricci flat metrics
  15. Ricci flat metrics: Explicit trajectories and rotational behaviour
  16. ...and 9 more sections

Key Result

Theorem A

On $CaP^2 \setminus \left\{ \text{ point } \right\},$$\mathbb{H}P^{m+1} \setminus \left\{ \text{ point } \right\}$ for $m \geq 1$ and on the vector bundle associated to $(G,H,K) = (Sp(m+1), Sp(1) \times Sp(m), U(1) \times Sp(m))$ for $m \geq 3,$ there exist a $1$-parameter family of non-homothetic c

Theorems & Definitions (77)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 1.1
  • Proposition 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 2.1
  • ...and 67 more