Mathematical Sciences: Applications of Variational Problems on Riemann Surfaces to Low Dimensional Geometry

9626565 Wolf建议的研究领域是变分和极小曲面。具体地说，提出了两个主题：在欧氏空间中寻找极小曲面的TeichMuller理论方法，以及通过曲面族调和映射族的双曲三维流形的几何。最小表面最早出现在比利时物理学家约瑟夫·普拉特于19世纪所做的一系列肥皂膜实验中。这些曲面具有面积最小化的性质：给定空间中的一条闭合轮廓，跨越该轮廓的最小曲面在所有边界相同轮廓的曲面中具有最小的曲面面积。由于这一性质和其他极端性质，极小曲面在物理、化学和材料研究中得到了广泛的应用。所提出的研究涉及使用复分析方法来构造这种极小曲面族，以及使用极小曲面技术来研究负曲三维空间。

9626565 Wolf The proposed research lies in the area of variational calculus and minimal surfaces. Specifically, two topics are proposed: Teichmuller theoretic methods for finding minimal surfaces in Euclidean 3-space, and the geometry of a hyperbolic three-manifold via families of harmonic maps of surfaces thereto. Minimal surfaces first arose in a series of soap film experiments done by the Belgian physicist Joseph Plateau in the 19th century. These surfaces have an area-minimizing property: given a closed contour in space, the minimal surface spanning it has the smallest surface area amongst all surfaces bounding the same contour. Because of this and other extremal properties minimal surfaces find many applications in physics, chemistry and materials research. The proposed research has to do with constructing families of such minimal surfaces using complex analysis methods and a study of negatively curved 3-spaces using minimal surface techniques.