Mean curvature flow, minimal surfaces and Ricci flow

点击翻译按钮获取中文摘要

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.