Mathematical Sciences: The Geometry of Surfaces in Three Dimensional Manifolds

9704286这个项目研究的是三维黎曼流形中的曲面几何。重点是等周问题和分段线性极小曲面理论。一个基本的等周问题需要一个包含给定体积对的最小面积曲面。双泡猜想断言，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。最近，这位研究人员与R.Schlafly合作，解决了这个猜想，解决了两个等量的问题，但一般情况下仍然是一个悬而未决的问题。曲面的分段线性理论考虑的是由平板粘合在一起的曲面，很像计算机图形学中构造的曲面。该项目考虑的问题涉及决定肥皂泡形状的自然界过程、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。曲面的分段线性理论可以应用于各种曲面的分类、计数和算法识别。

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 .