Mean curvature flow, minimal surfaces and Ricci flow

平均曲率流、最小曲面和里奇流

• 批准号: 1311795
• 负责人: Fernando Marques
• 金额: $ 15.79万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311795&HistoricalAwards=false
• 关键词: Mean; curvature; flow; minimal; surfaces

The three topics in the title of the project enjoy strong connections and analogies between them, which for some aspects are also directly useful in a rigorous setting. Thus, a large portion of the proposed project concerns the further extension of gluing techniques for solutions of the nonlinear partial differential equations of minimal surfaces and constant mean curvature surfaces, to study important existence and uniqueness questions for singularities and long time behavior of the two curvature flows. Such transplantation of techniques has already proven successful, also in the previous work of the P.I.Firstly, in the project the P.I. will, in collaboration with Prof. Nicos Kapouleas (at Brown University) and Dr. Stephen J. Kleene (at MIT), continue their joint work on gluing constructions for minimal surfaces and mean curvature flow singularities for surfaces in 3-manifolds, and applications to solitons in the curvature flows. Secondly, the PI will, in collaboration with Dr. Höskuldur P. Halldorsson, establish existence results for new types of long time solutions under mean curvature flow in Euclidean space. Thirdly, the P.I. will, alone or in collaboration with others, study time-dependent gluing constructions in mean curvature flow, and explore an analogous problem for 3-dimensional Ricci flow, which will further help guiding the on-going study of properties of this flow by many other researchers. Fourth, alone or in collaboration with others, initiate the study of gluing constructions for other nonlinear geometric PDEs. For example in relation to conformal geometry, complex geometry and constraints on Ricci curvature.Minimizing the area of an interface between two regions is a fundamental problem which arises in many scientific and engineering applications. Mean curvature flow, being defined as the fastest way to locally decrease the area of a given surface, is formulated as a partial differential equation, very similar in nature to that which governs the flow of heat in a material. While key foundational results are known, many of the basic questions remain unanswered. Already, many very striking applications, lauded by experts across the sciences, have in recent years followed from the study of these particular flows. Many more are expected to arise both within and outside of mathematics, such as in astronomy, to black holes and the large-scale structure of the universe, and in chemistry, to complex molecules. The proposal involves research problems at varying levels, and so there is rich opportunity and concrete plans to include as well undergraduate students as graduate students, postdoctoral and faculty researchers across institutions in the project. Given the manifest geometric and physically motivated nature of the material, the ideas and results of the project are also very suitable for an inspiring communication, visually and otherwise, to a broad audience.

该项目标题中的三个主题享有它们之间的牢固联系和类比，在某些方面，在严格的环境中也直接有用。这是拟议项目的很大一部分涉及针对最小表面和恒定平均曲率表面的非线性部分偏微分方程的胶合技术的进一步扩展，以研究重要的存在和独特的存在和两个曲率流动的奇异性和长时间行为。在P.I.先前的工作中，技术的移植已经证明是成功的。首先，在项目中威尔与尼科斯·卡普洛斯（Nicos Kapouleas）教授（在布朗大学）和斯蒂芬·J·克莱恩（Stephen J. Kleene）博士（在麻省理工学院）合作，继续他们在3个manifolds中的最小表面和平均曲率流量奇异性方面的粘合构造以及在曲率流中的固体中的应用。其次，PI将与HöskuldurP. Halldorsson博士合作，为欧几里得空间中平均曲率流下的新型长期解决方案建立了存在的结果。第三，P.I.将单独或与他人合作，研究平均曲率流中的时间依赖性的吸血结构，并探索三维RICCI流动的类似问题，这将进一步帮助指导许多其他许多研究人员对这种流量的持续研究。第四，单独或与他人合作，启动了其他非线性几何PDE的糖基化研究。例如，与保形几何形状，复杂的几何形状和RICCI曲率的约束有关。将两个区域之间的界面区域量化是一个基本问题，在许多科学和工程应用中都会出现。平均曲率流（被定义为局部降低给定表面面积的最快方法）被配制为偏微分方程，与控制材料中热量流动的性质非常相似。虽然关键的基础结果是已知的，但许多基本问题仍然没有解决。近年来，从研究这些特定流动的研究之后，许多非常引人注目的应用受到了科学专家的称赞。预计将出现更多的数学内部和外部，例如天文学，黑洞和宇宙的大规模结构以及化学的大规模结构。该提案涉及不同级别的研究问题，因此，在该项目的机构中，有丰富的机会和具体计划，以及本科生，研究生，博士后和教职员工研究人员。鉴于材料的明显几何形状且具有物理动机的性质，该项目的思想和结果也非常适合于视觉上和其他人的启发性沟通。