Cohomogeneity one Ricci solitons

Thesis

In this work we study the cohomogeneity one Ricci soliton equation viewed as a dynamical system. We are particularly interested in the relation between integrability of the associated system and the existence of explicit, closed form solutions of the soliton equation. The contents are organized as follows.

The first chapter is an introduction to Ricci ow and Ricci solitons and their basic properties. We reformulate the rotationally symmetric Ricci soliton equation on ℝⁿ⁺¹ as a system of ODE's following the treatment in [14].

In Chapter 2 we carry out a Painlevé analysis of the previous system. For the steady case, dimensions where n is a perfect square are singled out. The cases n = 4, 9 are particularly distinguished. In the expanding case, only dimension n = 1 is singled out.

In Chapter 3 we reformulate the cohomogeneity one Ricci soliton equation as a Hamiltonian system with constraint. We obtain a conserved quantity for this system and produce explicit formulas for solitons of dimension 5.

In Chapter 4 we introduce the notion of superpotential and use it to produce more explicit formulas for solitons of steady, expanding and shrinking type.

In Chapter 5 we carry out a Painlevé anaylsis of the Hamiltonian corresponding to solitons over warped products of Einstein manifolds with positive scalar curvature. This analysis singles out the cases discussed in the previous chapters. We also carry out an analysis of the Hamiltonian corresponding to the Bérard Bergery ansatz [5]. This analysis singles out a 1-parameter family of solutions.

Authors: de la Parra, A

• DOI: 10.5287/ora-6qpyo0rrd
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford