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

数学科学：黎曼曲面变分问题在低维几何中的应用

• 批准号: 9626565
• 负责人: Michael Wolf
• 金额: $ 6万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626565&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applications; Variational; Problems

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.

小行星9626565 建议的研究在于变分和极小曲面领域。具体而言，提出了两个主题：Teichmuller理论的方法，寻找最小的表面在欧几里德3-空间，和几何的双曲三流形通过家庭的调和映射的表面。 最小表面最早出现在世纪比利时物理学家约瑟夫·普拉托（Joseph Plateau）的一系列肥皂膜实验中。这些曲面具有面积最小化属性：给定空间中的闭合轮廓，跨越它的最小曲面在所有边界相同轮廓的曲面中具有最小的表面积。由于这一点和其他极端性质，极小曲面在物理、化学和材料研究中有许多应用。建议的研究与建设家庭的这种极小曲面，使用复杂的分析方法和研究负弯曲的3-空间，使用极小曲面技术。