Mathematical Sciences: The Global Geometry of Extremal Surfaces

数学科学：极值曲面的整体几何

• 批准号: 8908064
• 负责人: Robert Kusner
• 金额: $ 3.03万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8908064&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Global; Geometry; Extremal

The principal investigator will study the global differential geometry of surfaces in three dimensional manifolds. The research focuses on surfaces which are extrema for natural symmetric variational problems arising in mathematics and physics. Computable geometric moduli for extremal surfaces will be developed. The robustness of this theory will be investigated by relaxing some symmetry assumptions on the ambient space. And these methods will be extended to other geometric variational problems with symmetry or approximate symmetry. Three-dimensional space is an example of a three-manifold. If infinity is collapsed to one point, this space becomes the three-dimensional sphere. This sphere has constant sectional curvature because relative to every plane at every point, the sphere bends the same. The principal investigator will study deep problems concerning the minimality of surfaces which lie in such three-manifolds. Soap films are familiar examples of such minimal surfaces.

主要研究人员将研究三维流形中曲面的整体微分几何。研究的重点是数学和物理中出现的自然对称变分问题的极值曲面。将开发用于极值曲面的可计算几何模数。这一理论的稳健性将通过放松环境空间上的一些对称性假设来研究。并将这些方法推广到其他对称或近似对称的几何变分问题。三维空间是三维流形的一个例子。如果无限空间塌陷到一点，这个空间就变成了三维球体。该球体具有恒定的截面曲率，因为相对于每个点上的每个平面，球体的弯曲是相同的。首席研究员将研究与这种三维流形中的曲面的极小性有关的深层次问题。肥皂膜是这种极小表面的常见例子。