Asymptotic Behaviour of Geometric Flows

几何分析领域一直很有影响力，产生了大量的重要成果，这些成果不仅在这一研究领域留下了自己的印记，而且也推动了数学的其他领域。这一点以该领域的一个基本问题的研究为例，即所谓的高原问题，即寻找跨越给定边界曲线的最小表面积的曲面。为解决这一问题而开发的技术在现代分析和变分领域的发展中至关重要，例如勒贝格的积分理论。最近，佩雷尔曼用利玛窦流动证明了著名的庞加莱猜想，证明了这一点。几何分析领域的一个重要研究课题是研究自然几何泛函，如流形之间或曲面面积之间的映射的Dirichlet能量，以及在上述情况下的临界点、调和映射和极小曲面。产生这些临界点的一种自然方法是通过梯度下降的方法将初始对象流动到临界对象。这种方法是由Eells和Sampson于20世纪60年代在狄利克雷特能量的背景下引入的，导致了调和映射流的定义，现在是几何分析和更广泛的数学中许多其他泛函的通用方法。理解几何梯度流的一个关键性质是渐近行为。通常，通过紧致性论证可以获得第一个收敛结果，但这只适用于一系列时间，并且将其升级到完全收敛通常要困难得多。在研究这种收敛的许多环境中，最有力的结果是一个估计，称为Lojasiewicz-Simon不等式，它确保在临界点附近流动的良好行为。此外，这些估计通常提供关于流的收敛速度的先验界限，这在应用环境中尤其相关，其中流被用来对物理系统进行建模，并且需要精确的数值模拟。除了它们在梯度流中的应用，Lojasiewicz-Simon估计在获得泛函能量谱的结果方面也是有用的，例如排除聚集点。继Simon在20世纪80年代的开创性工作之后，Lojasiewicz-Simon估计已成功地应用于几何分析和其他领域的各种环境中，包括控制理论和数值优化。不幸的是，Simon开创的方法不能扩展到奇点形成的环境，例如当拓扑在极限中改变时，这是一个重大的挫折，因为在许多感兴趣的环境中，奇点可以并且确实会发生。有鉴于此，这个项目的主要目标是在不符合Simon原始方法的情况下推导Lojasiewicz-Simon估计，并探索在存在奇点时几何流动收敛的应用。我们将集中讨论Dirichlet能量，其中原始调和映射流和由Ding，Li和Liu首先在一种特殊情况下引入并由Rupflin和Topping推广的变量通常都形成奇点。就目前而言，关于奇异环境下Lojasiewicz-Simon不等式的已知结果很少，而且这些结果大多是最近的，所以这种方法具有高度的新颖性。因此，这些方法在几何分析的范围之外具有重大的潜在影响。上述项目属于EPSRC数学分析研究领域，因为关键技术和理论来自于偏微分方程组的分析。还有与EPSRC几何和拓扑学研究领域的链接，因为这些偏微分方程自然起源于微分和黎曼几何，因此关于它们的全局行为和沿途发展的技术的结果在这些领域产生了影响。

The field of geometric analysis has been very influential, producing a wealth of important results which have not only left their mark on this research area but have pushed forwards other areas of mathematics too. This is exemplified by the study of one of the foundational problems of the field, the so called Plateau problem of finding surfaces of least surface area spanning a given boundary curve. The techniques that were developed to solve this were crucial in the development of the fields of modern analysis and the calculus of variations, for example Lebesgue's theory of integration. More recently, this is seen in Perelman's proof of the famous Poincaré conjecture using Ricci flow. A key topic of research in the field of geometric analysis is the study of natural geometric functionals, such as the Dirichlet energy of maps between manifolds or the area of a surface, and their critical points, harmonic maps and minimal surfaces in the above cases. A natural way of producing these critical points is to flow an initial object to a critical one by means of gradient descent. This approach was introduced by Eells and Sampson in the 1960s in the context of the Dirichlet energy, leading to the definition of harmonic map flow, and is now a common approach for many other functionals in geometric analysis and mathematics more generally. One of the key properties to understand about a geometric gradient flow is the asymptotic behaviour. It is common that a first convergence result can be obtained via a compactness argument, but this will only apply along a sequence of times and upgrading this to full convergence is often much harder. The most powerful result in many settings for studying this convergence is an estimate, called a Lojasiewicz-Simon inequality, which ensures good behaviour of the flow near critical points. In addition, these estimates often provide a priori bounds on the rate of convergence of the flow, which is relevant in particular in applied settings where the flow is being used to model a physical system and an accurate numerical simulation is desired. Alongside their applications to gradient flows, Lojasiewicz-Simon estimates are useful in obtaining results on the energy spectrum of the functional, such as to exclude accumulation points. Following on from the seminal work of Simon in the 1980s, Lojasiewicz-Simon estimates have been successfully applied in diverse settings in geometric analysis and beyond, including in control theory and numerical optimisation. Unfortunately, the approach pioneered by Simon does not extend to settings where a singularity forms, for example when the topology changes in the limit, which is a major setback as in many settings of interest, singularities can and do occur. In light of this, the key aim of this project is to derive Lojasiewicz-Simon estimates in situations not amenable to Simon's original method and to explore applications to the convergence of geometric flows in the presence of singularities. We will focus on the Dirichlet energy where both the original harmonic map flow and a variant, introduced first in a special case by Ding, Li and Liu and generalised by Rupflin and Topping, are known to form singularities in general. As it stands, there are only a few known results on Lojasiewicz-Simon inequalities in singular settings, and these are mostly very recent, and so this approach has a high degree of novelty. As a result, there is a significant potential impact of these methods beyond the confines of geometric analysis.The project outlined above falls within the EPSRC Mathematical Analysis research area as the key techniques and theory come from the analysis of PDEs. There are also links to the EPSRC Geometry and Topology research area since these PDEs originate naturally in differential and Riemannian geometry and so the results on their global behaviour and techniques developed along the way have consequences in these fields.