Taub-Bolt and Taub-NUT solutions and their behaviour under the Ricci flow

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Context:Taub-Bolt and Taub-NUT solutions are a subclass of Ricci flat metrics on a manifold, and can appear on manifolds of different dimensions. These metrics appear in physics in the study of Black Holes. However here we are talking about Euclidean black holes since a metric is used rather than a Lorentzian metric that is used in General Relativity. Maybe the thesis would be of use to physicists.Details of the Thesis:In dimension four and assuming certain symmetries gives rise to a class of Taub-Bolt and Taub-NUT solutions. The Bolt and the NUT refer to the topology of the manifold. The Taub-Bolt solutions are metrics on two-dimensional complex projective space minus a four-dimensional ball, making it a manifold with boundary, whereas the Taub-NUT solutions are on the closed four-dimensional ball. The Taub-Bolt solutions can be stable or unstable under the Ricci flow, while the Taub-NUT solutions are stable. There has been some work done by Holzegel, Schmelzer and Warnick that suggests that if we perturb the unstable Taub-Bolt solutions, under the Ricci flow, the metric will flow to either a stable Taub-Bolt or the Taub-NUT solution. As mentioned before, the Taub-NUT solution however is on a different manifold than that of the Taub-Bolt solutions. The Ricci flow must therefore be changed to Ricci flow with surgery. This involves following the Ricci flow for a period of time, pausing to change the topology (perform surgery), then continuing the Ricci flow on the different manifold. The use of surgery is used because the Ricci flow has a finite blow up time. Hamilton and Perelman came up with Ricci flow with surgery to continue the flow beyond what is normally possible.The study of the flow beyond stability questions so far is via computational methods. The question of long-time existence and convergence of the flow has not yet been answered. The thesis would be to answer this via analysis of the partial differential equations that arise (i.e the Ricci flow equation). More generally, I would explore the behaviour of perturbations of Taub-Bolt and Taub-NUT solutions via analytic means.This project falls within the EPSRC Geometry and Topology research area.