Asymptotic Behaviour of Geometric Flows

该领域的几何分析一直是非常有影响力的，产生了丰富的重要成果，不仅留下了自己的印记，这一研究领域，但也推动了其他领域的数学。这是例证的研究领域的基础问题之一，所谓的高原问题，寻找表面的最小表面积跨越给定的边界曲线。该技术的发展，以解决这是至关重要的发展领域的现代分析和变分法，例如勒贝格的理论整合。最近，佩雷尔曼使用里奇流证明了著名的庞加莱猜想。几何分析领域的一个关键研究课题是自然几何泛函的研究，例如流形之间映射或曲面面积的Dirichlet能量，以及它们的临界点，调和映射和极小曲面在上述情况下。产生这些临界点的一种自然方法是通过梯度下降将初始对象流到临界对象。这种方法是由Eells和Sampson在1960年代在Dirichlet能量的背景下引入的，导致了调和映射流的定义，现在是几何分析和数学中许多其他泛函的常用方法。了解几何梯度流的关键性质之一是渐近行为。通常，可以通过紧性参数获得第一个收敛结果，但这将仅适用于沿着时间序列，并且将其升级为完全收敛通常要困难得多。在许多研究这种收敛性的设置中，最强大的结果是一个估计，称为Lojasiewicz-Simon不等式，它确保了临界点附近的流动的良好行为。此外，这些估计通常提供关于流的收敛速率的先验界限，这在应用设置中特别相关，其中流被用于对物理系统进行建模并且期望精确的数值模拟。除了它们在梯度流中的应用，Lojasiewicz-Simon估计在获得泛函能谱的结果方面是有用的，例如排除聚集点。继西蒙在20世纪80年代的开创性工作之后，Lojasiewicz-Simon估计已成功应用于几何分析及其他领域的各种环境，包括控制理论和数值优化。不幸的是，西蒙开创的方法没有扩展到奇点形成的设置，例如当拓扑在极限中变化时，这是一个重大的挫折，因为在许多感兴趣的设置中，奇点可以并且确实发生。鉴于此，该项目的主要目的是获得Lojasiewicz-Simon估计的情况下，不适合西蒙的原始方法，并探讨应用程序的收敛性的几何流中存在的奇点。我们将专注于狄利克雷能源的原始调和映射流和一个变种，介绍了第一次在特殊情况下丁，李和刘和推广的Rupflin和Topping，是已知的一般形成奇点。就目前而言，关于奇异情形下Lojasiewicz-Simon不等式的已知结果很少，而且这些结果大多是最近才得到的，因此这种方法具有很高的新奇。因此，这些方法的潜在影响超出了几何分析的范围。上述项目福尔斯属于EPSRC数学分析研究领域，因为关键技术和理论来自偏微分方程的分析。也有链接到EPSRC几何和拓扑研究领域，因为这些偏微分方程自然起源于微分和黎曼几何，因此其全球行为和技术的结果沿着发展的方式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.