Advances in Mean Curvature Flow: Theory and Applications

点击翻译按钮获取中文摘要

This project aims to develop the theoretical framework of the singularity formation of the Mean Curvature Flow. The Mean Curvature Flow is a geometric flow that describes the motion of a surface. It was introduced by Mullins as a model for the formation of grain boundaries in annealing metals. It also appears as the flow to equilibrium of soap films, the motion of embedded branes in approximations of the renormalisation group flow in theoretical physics, boundaries of Ginzburg-Landau equations of simplified superconductivity and as a method of denoising in image processing. The results of this project, as well as the methods pioneered, will enable the next generation of applications.This proposal lies in the intersection of the EPSRC research areas Mathematical Analysis and Geometry and Topology with applications to Mathematical Physics and Algebra. It has underpinning relevance ranging from fundamental problems in theoretical physics to current issues in engineering. At the heart of these problems is a single system of geometric Partial Differential Equations (PDE). Such equations have had a tremendous impact in mathematics: they have been extremely successful with applications over diverse areas such as topology (Poincaré Conjecture, Geometrisation conjecture), Kähler geometry (minimal model problem), gravitation (Penrose inequality), image processing and material science (Martensite, nonlinear plate models).Geometric PDE are inherently nonlinear and therefore singularities are expected to occur. In fact, these singularities turn out to be useful - they tell us something about the underlying geometry of our object. This project will develop our understanding of singularities of systems of geometric PDE, an extremely important area in geometry, analysis and PDE theory which is relatively poorly understood. In principle, the singularities of systems of nonlinear partial differential equations may be unstructured, but due to their geometric origins, the singularities of systems of geometric PDE display a surprising order. This project seeks to characterise the mechanisms of the formation of singularities, obtain classifications of the singularity models, and to develop a geometric hierarchy of singularity models and quantitatively analyse their stability. Modern applications of geometric flows require a detailed understanding of singularities using an integrated approach combining algebra, analysis, geometry and topology.More precisely, the proposed research consists of the following themes: understanding the singularity formation of the Mean Curvature Flow in high codimension and in curved background spaces, developing new concentration compactness results to analyse the singularities and surgery procedures to geometrically undo the singularity formation, and finally exploring applications of the new theory to various fields of mathematics. The results pioneered in this project will have a direct and significant impact on geometry, analysis, and topology. Furthermore, the methodologies and techniques developed in this project can also be applied to a number of outstanding problems in physics, biology, engineering and computer imaging.