Asymptotic Behaviour of Geometric Flows

几何分析领域的影响非常大，产生了大量重要的结果，这些结果不仅在这个研究领域留下了自己的印记，而且也推动了数学其他领域的发展。该领域的一个基础问题，即所谓的求跨越给定边界曲线的最小表面积曲面的高原问题，就是一个例证。为了解决这个问题而开发的技术在现代分析和变分学领域的发展中至关重要，例如勒贝格的积分理论。最近，佩雷尔曼用里奇流证明了著名的庞加莱猜想。几何分析领域的一个重要研究课题是自然几何泛函的研究，如流形之间映射或曲面面积之间映射的狄利克雷能量及其临界点、调和映射和极小曲面。产生这些临界点的一种自然方法是用梯度下降的方法使一个初始物体流向临界点。这种方法是由Eells和Sampson在20世纪60年代在Dirichlet能量的背景下引入的，导致了谐波映射流的定义，现在是几何分析和数学中许多其他泛函的常用方法。理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，但这只适用于时间序列，并且将其升级为完全收敛通常要困难得多。在研究这种收敛性的许多设置中，最有力的结果是一个估计，称为Lojasiewicz-Simon不等式，它确保了在临界点附近的流的良好行为。此外，这些估计通常提供了流动收敛速度的先验界限，这在应用环境中尤其相关，其中流动被用来模拟物理系统，并且需要精确的数值模拟。除了将Lojasiewicz-Simon估计应用于梯度流之外，它还有助于获得函数能谱的结果，例如排除累积点。继Simon在20世纪80年代的开创性工作之后，Lojasiewicz-Simon估计已经成功地应用于几何分析和其他领域的各种设置，包括控制理论和数值优化。不幸的是，Simon开创的方法并没有扩展到奇点形成的环境中，例如当拓扑在极限处发生变化时，这是一个主要的挫折，因为在许多感兴趣的环境中，奇点可以而且确实发生了。鉴于此，本项目的主要目的是在不适合Simon原始方法的情况下推导Lojasiewicz-Simon估计，并探索在奇点存在时几何流收敛的应用。我们将重点关注狄利克雷能量，在狄利克雷能量中，原始谐波映射流和一种变体，首先由Ding、Li和Liu在特殊情况下引入，并由Rupflin和Topping推广，已知通常会形成奇点。就目前而言，只有少数已知的结果是关于奇异环境下的Lojasiewicz-Simon不等式的，而且这些结果大多是最近才出现的，所以这种方法具有很高的新颖性。因此，这些方法在几何分析范围之外具有重要的潜在影响。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对偏微分方程的分析。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地起源于微分和黎曼几何，所以在它们的整体行为和技术发展的结果在这些领域的影响。

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.