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

背景：Taub-Bolt和Taub-NUT解决方案是流形上的Ricci平坦度量的子类，可以出现在不同维度的流形上。这些度量值出现在研究黑洞的物理学中。然而，我们在这里谈论的是欧几里得黑洞，因为我们使用的是度规，而不是广义相对论中使用的洛伦兹度规。也许这篇论文会对物理学家有所帮助。论文细节：在四维空间中，假设某些对称性产生一类Taub-Bolt解和Taub-NUT解。螺栓和螺母指的是歧管的拓扑。Taub-Bolt解是二维复射影空间上的度量，减去一个四维球，使它成为一个有边界的流形，而Taub-NUT解是在闭四维球上的。在Ricci流下，Taub-Bolt解可以是稳定的，也可以是不稳定的，而Taub-NUT解是稳定的。Holzegel，Schmelzer和Warnick的一些工作表明，如果我们扰动不稳定的Taub-Bolt解，在Ricci流下，度规将流向稳定的Taub-Bolt解或Taub-NUT解。如前所述，Taub-NUT解决方案与Taub-Bolt解决方案位于不同的流形上。因此，瑞奇血流必须通过手术改变为瑞奇血流。这涉及到跟随Ricci流一段时间，暂停以改变拓扑(执行手术)，然后在不同的流形上继续Ricci流。使用外科手术是因为Ricci流有一个有限的爆炸时间。汉密尔顿和佩雷尔曼提出了Ricci Flow，通过手术使流动继续超出正常可能的范围。到目前为止，对超越稳定性问题的流动的研究是通过计算方法进行的。这股潮流的长期存在和汇聚的问题尚未得到回答。本文将通过对由此产生的偏微分方程(即Ricci流动方程)的分析来回答这个问题。更广泛地说，我将通过解析方法探索Taub-Bolt和Taub-NUT解的扰动行为。这个项目属于EPSRC几何和拓扑学研究领域。

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.