Advances in Mean Curvature Flow: Theory and Applications

本计画旨在发展平均曲率流奇点形成的理论架构。平均曲率流是描述曲面运动的几何流。它是由Mullins引入的，作为退火金属中晶界形成的模型。它也出现在肥皂膜的平衡流、理论物理学中重正化群流近似的嵌入膜运动、简化超导的金斯堡-朗道方程的边界以及图像处理中的去噪方法中。该项目的结果，以及开创的方法，将使下一代的applications.This建议在于在EPSRC的研究领域的数学分析和几何和拓扑与应用数学物理和代数的交集。从理论物理学的基本问题到工程学的当前问题，它都具有基础性的相关性。在这些问题的核心是一个单一的系统的几何偏微分方程（PDE）。这类方程在数学中有着巨大的影响：它们在不同的领域都有着非常成功的应用，如拓扑学（庞加莱猜想，几何化猜想），凯勒几何（最小模型问题），引力（彭罗斯不等式），图像处理和材料科学（马氏体，非线性板模型）。几何偏微分方程本质上是非线性的，因此奇异性会出现。事实上，这些奇点被证明是有用的--它们告诉我们一些关于我们物体的基本几何形状的信息。这个项目将发展我们对几何偏微分方程系统奇点的理解，这是几何、分析和偏微分方程理论中一个非常重要的领域，人们对它的理解相对较少。原则上，非线性偏微分方程组的奇异性可能是非结构化的，但由于其几何起源，几何PDE系统的奇异性显示出令人惊讶的顺序。本项目旨在阐明奇点形成的机制，获得奇点模型的分类，并建立奇点模型的几何层次结构，并定量分析其稳定性。几何流的现代应用需要使用结合代数，分析，几何和拓扑的综合方法详细了解奇点。更确切地说，拟议的研究包括以下主题：理解高余维和弯曲背景空间中平均曲率流的奇异性形成，开发新的浓度紧凑性结果以分析奇点和外科手术程序以几何地消除奇点形成，最后探讨新理论在数学各个领域的应用。在这个项目中开创的结果将对几何，分析和拓扑结构产生直接和重大的影响。此外，在这个项目中开发的方法和技术也可以应用于物理学，生物学，工程和计算机成像中的一些突出问题。

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.