Variational Problems in Low Dimensional Geometry and Topology

9704949 Kusner这个项目处理各种几何变分问题。该项目的一个重点是几何模问题，特别是对于完全常数平均曲率曲面和共形平坦常数标量曲率度量。来自力学和辛几何的工具——在曲面的情况下，涉及旋量和四元数形式的Teichmuller理论方法和表示——将被用来理解这些模空间的结构。几何变分问题包括在满足一定边界或初始条件的目标族中寻找极值或最优目标的各种问题。例如，跨越给定边界的最小曲面将跨越同一边界的所有曲面的表面积最小化。最小表面，更一般地说，恒定平均曲率表面，可以用来模拟某些凝聚态物质-这是基于观察到，当两种均匀介质相遇时，它们的界面形成一个恒定平均曲率表面。

9704949 Kusner This project deals with various geometric variational problems. One focus of the project is geometric moduli problems, especially for complete constant mean curvature surfaces and conformally flat constant scalar curvature metrics. Tools from mechanics and symplectic geometry - and in the case of surfaces, Teichmuller theoretic methods and representations involving spinors and quaternionic forms - is to be used to understand the structure of these moduli spaces. Geometric variational problems include a variety of problems where finding extremal or optimal objects amongst families of objects all satisfying certain boundary or initial conditions. For example, a minimal surface spanning a given boundary minimizes the surface area amongst all surfaces spanning the same boundary. Minimal surfaces, and more generally, constant mean curvature surfaces, can be used to model certain condensed matter - this is based on the observation that when two homogeneous media meet their interface forms a constant mean curvature surface.