Loss of compactness phenomena for Dirichlet energy

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Throughout the history of mathematics, the link between geometry and analysis has given insight into both fields and led to many new developments, in turn solving major problems, including the Plateau problem and the Poincaré conjecture. Following these results, research into the interplay between these fields remains active, with many open problems.An important area of geometric analysis is the study of geometric problems in the calculus of variations, as many geometric objects can be defined as critical points of suitably defined functionals. Two examples of naturally occurring functionals are the Dirichlet energy of maps between manifolds and the area of surfaces, whose critical points are harmonic maps and minimal surfaces, respectively. One problem in the calculus of variations is proving the existence of critical points. Two important methods to do this are evolution by gradient flow and directly using minimising sequences. In both cases, one aims to preserve topological properties; however, this requires a successful compactness theory that can prove a major hurdle. In the context of Dirichlet energy, the gradient flow was introduced by Eells-Sampson in the 1960s, who proved the existence of globally defined smooth solutions for negatively curved targets. However, the work of Eells-Wood in the 1970s showed that, in general, one must expect the formation of singularities.Subsequently, this question of loss of compactness has inspired research into what type of convergence can be expected to hold. In 1985, Struwe showed that solutions to the harmonic map flow were smooth away from at most finitely many singularities caused by a concentration of energy. Further research into the behaviour of almost harmonic maps has yielded a compactness theory proving strong convergence away from finitely many points where bubbles form and establishing "bubble tree convergence," where the loss of energy is accounted for by the bubbles.Despite these results, there are still many open problems concerning the optimal description of the flow near singularities and the optimal compactness theory for such a subsequence, such as the quantitative relationship between the blow-up scale and the decay of the tangent space. In the past few years, there has been progress on answering questions on the fine structure of almost harmonic map flows, such as the work of Del-Pino et al. on the location and rate of formation of singularities. To date, the fine analysis of loss of compactness phenomena remains an open problem with many challenges. This project aims to contribute to a better understanding of these questions, in particular the fine structure of the energy landscape and its impact near critical points and on the dynamics of gradient flows. In order to investigate this, we will firstly consider highly symmetric settings, where singularities are known to occur, aiming to obtain optimal results for this particular setting and use this case to test methods developed to study more general settings. To do so, we will look to use a combination of techniques from modern mathematics such as Teichmuller theory and Lojasiewicz estimates, together with explicit examples using the recent works by Rupflin as inspiration. Furthermore, we expect to develop novel techniques that would be applicable to other non-linear PDEs and, hence, be of interest to the wilder field of mathematics.The core part of this research project falls within the EPSRC Mathematical Analysis research area, as we aim to use and develop techniques from functional analysis as well as elliptic and parabolic PDE theory. There are also multiple connections to the EPSRC Geometry and Topology research area, as loss of compactness phenomena are closely connected to topological phenomena and degeneracy of geometric structure, and as the problem we intend to study is inspired by geometry.