Asymptotic Behaviour of Geometric Flows

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.