Mathematical Sciences: The Geometry of Surfaces in Three Dimensional Manifolds

数学科学：三维流形中的曲面几何

• 批准号: 9704286
• 负责人: Joel Hass
• 金额: $ 10.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704286&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Surfaces; Three

9704286 Hass This project deals with the geometry of surfaces in three dimensional Riemannian manifolds. The focus is on isoperimetric problems and the theory of piecewise-linear minimal surfaces. A basic isoperimetric problem asks for the smallest area surface enclosing a given pair of volumes. The Double Bubble Conjecture asserts that the shape assumed by a compound soap bubble enclosing two regions answers this question. This conjecture was recently solved by the investigator, in collaboration with R. Schlafly, for two equal volumes, but the general case remains an open problem. The piecewise-linear theory of surfaces considers surfaces which are glued together from flat pieces, much like the surfaces constructed in computer graphics. The problems considered in this project relate to processes in nature which determine the shape of soap bubbles, the erosion of boulders, the melting of ice, crystal growth, flame propagation, image enhancement and the evaporation of puddles. The piecewise-linear theory of surfaces can be applied to the classification, enumeration and algorithmic recognition of various surfaces .

9704286 HASS这个项目涉及三维Riemannian歧管的表面几何形状。重点是等等问题和分段线性最小表面的理论。一个基本的等速度问题要求封闭给定体积的最小区域表面。双重气泡猜想断言，封闭两个区域的化合物肥皂泡假定的形状回答了这个问题。该猜想最近由研究人员与R. Schlafly合作解决了两个相等的批量，但总体情况仍然是一个空旷的问题。表面的分段线性理论认为表面是从平坦的碎片中粘合在一起的，就像计算机图形中构建的表面一样。 该项目中考虑的问题与自然界的过程有关，这些过程决定了肥皂泡的形状，巨石的侵蚀，冰的融化，晶体生长，火焰传播，图像增强和水坑的蒸发。表面的分段线性理论可以应用于各种表面的分类，枚举和算法识别。