Mathematical Sciences: Complete Surfaces in Three-Space

数学科学：三空间中的完备曲面

• 批准号: 8702383
• 负责人: Brian Smyth
• 金额: $ 13.13万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702383&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Complete; Surfaces; Three

Brian Smyth and Frederico Xavier will continue their work on the theory of complete minimal surfaces. This is a subject where great advances have taken place in the last decade. These were to a large extent based on the conformal character of such surfaces. In the current project attention will be focussed on more general surfaces. Many of the important unresolved problems in this area depend on an understanding of the analytic properties of the Gauss map. In particular it is important to investigate which meromorphic functions on the unit disc arise as Gauss maps of complete minimal surfaces. One of the principal motivations for this work arises from Efimov's result that any complete surface in three dimensional space which has negative Gauss curvature must have points where this curvature is arbitrarily close to zero. This in turn is related to the conjecture of Caratheodory that every compact surface in three dimensional space which has positive Gauss curvature must have at least two umbilic points. Smyth and Xavier have already made substantial progress towards this conjecture and will continue their efforts in this direction. Other problems which will be investigated include the question as to whether there are bounded non-compact complete surfaces with negative Gauss curvature.

布莱恩·史密斯和弗雷德里克·泽维尔将继续他们的工作， 完备极小曲面理论这是一个主题， 在过去十年中取得了巨大的进步。这些是为了 很大程度上基于这种表面的共形特性。 在目前的项目中，注意力将集中在更一般的 表面。这一领域许多重要的未解决问题 依赖于对分析性质的理解， 高斯映射。特别重要的是要调查哪些 单位圆盘上的亚纯函数作为高斯映射出现， 完备极小曲面 这项工作的主要动机之一来自于 Efimov的一个结果，即三维空间中的任何完整曲面 一个具有负高斯曲率的空间一定有这样的点， 该曲率任意地接近于零。这进而 与Caratheodory的猜想有关， 三维空间中具有正高斯的曲面 曲率必须至少有两个脐点。史密斯和泽维尔 已经在这个猜想上取得了实质性的进展 并会继续朝这个方向努力。其他问题 将被调查的问题包括， 有界非紧完备曲面 高斯曲率。