Mean curvature flow, minimal surfaces and Ricci flow

项目标题中的三个主题之间有很强的联系和类比，在某些方面，这在严格的环境中也是直接有用的。因此，很大一部分的建议项目涉及进一步扩展的胶合技术的解决方案的非线性偏微分方程的最小曲面和常平均曲率曲面，研究重要的存在性和唯一性问题的奇异性和长时间的行为的两个曲率流。这种技术的移植已经被证明是成功的，在P.I.以前的工作中也是如此。将与Nicos Kapouleas教授（布朗大学）和Stephen J. Kleene博士（麻省理工学院）合作，继续他们在最小曲面和3流形中曲面的平均曲率流奇点的胶合构造以及曲率流中孤子的应用方面的联合工作。其次，PI将与Höskuldur P. Halldorsson博士合作，建立欧氏空间中平均曲率流下新型长时间解的存在性结果。第三，P.I.将单独或与他人合作研究平均曲率流中的时间相关胶合结构，并探索三维Ricci流的类似问题，这将有助于进一步指导许多其他研究人员正在进行的这种流的性质的研究。第四，单独或与他人合作，开展其他非线性几何偏微分方程胶合构造的研究。例如，在保形几何、复几何和Ricci曲率的约束中，最小化两个区域之间的界面面积是许多科学和工程应用中出现的一个基本问题。平均曲率流被定义为局部减小给定表面面积的最快方式，被公式化为偏微分方程，其性质与控制材料中的热流的性质非常相似。虽然关键的基础结果是已知的，但许多基本问题仍然没有答案。近年来，对这些特殊流动的研究已经产生了许多非常引人注目的应用，这些应用受到了科学界专家的称赞。在数学的内部和外部，预计还会有更多的问题出现，例如在天文学中，黑洞和宇宙的大尺度结构，以及在化学中，复杂分子。该提案涉及不同层次的研究问题，因此有丰富的机会和具体的计划，包括本科生以及研究生，博士后和跨机构的研究人员在项目中。鉴于材料明显的几何和物理动机性质，该项目的想法和结果也非常适合在视觉上和其他方面与广大观众进行鼓舞人心的交流。

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.