Mathematical Sciences: Some Problems in Geometric Analysis

数学科学：几何分析中的一些问题

• 批准号: 8702871
• 负责人: Paul Yang
• 金额: $ 7.6万
• 依托单位: University of Southern California
• 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: 1987-06-15 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702871&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Problems; Geometric

Paul Yang will work on three central problems in geometric analysis. The first of these involves investigating which functions on the sphere are Gaussian curvature functions of metrics conformally equivalent to the standard metric. The second is to find out to what extent the universal cover of certain surfaces of negative curvature resembles the ball. In the final question the objective is to study the analytic and geometric nature of regions which give reliable data when used as sampling regions. The Nirenberg problem concerning characterizing Gaussian curvature functions is a delicate one in view of the necessary condition observed by Kazdan and Warner. Yang will focus his efforts on an investigation of functions admitting only two critical points. In the context of the uniformization problem Yang will investigate the conjecture that the universal cover of a Kahler manifold of negative sectional curvature is biholomorphically equivalent to a bounded domain in complex n-space. The Pompieu problem will be reformulated in terms of the existence of an infinite sequence of eigenfunctions to two dual systems of overdetermined boundary conditions.

Paul Yang 将致力于解决几何分析中的三个核心问题。第一个涉及研究球体上的哪些函数是与标准度量共形等价的度量的高斯曲率函数。第二个是找出某些负曲率表面的通用覆盖层与球的相似程度。在最后一个问题中，目标是研究区域的分析和几何性质，这些区域在用作采样区域时提供可靠的数据。 鉴于 Kazdan 和 Warner 观察到的必要条件，关于表征高斯曲率函数的尼伦伯格问题是一个微妙的问题。杨将集中精力研究只承认两个关键点的函数。在均匀化问题的背景下，杨将研究负截面曲率的卡勒流形的通用覆盖双全纯等价于复 n 空间中的有界域的猜想。庞皮厄问题将根据两个超定边界条件的对偶系统存在无限序列的本征函数来重新表述。