Mathematical Sciences: The Geometry of Surfaces in Three Dimensional Manifolds

9704286 Hass这个项目处理三维黎曼流形中的曲面几何。重点是等周问题和分段线性最小曲面理论。一个基本的等周问题要求给定一对体积的最小面积表面。“双泡猜想”断言，一个包围两个区域的复合肥皂泡所呈现的形状回答了这个问题。这个猜想最近被研究者与R. Schlafly合作解决了，在两个相等的卷中，但一般情况下仍然是一个开放的问题。表面的分段线性理论考虑的是由平面粘合在一起的表面，很像计算机图形学中的表面结构。在这个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、晶体的生长、火焰的传播、图像的增强和水坑的蒸发。曲面的分段线性理论可以应用于各种曲面的分类、枚举和算法识别。

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 .