Analytic Problems in Geometry

几何解析问题

• 批准号: 9706507
• 负责人: Paul Yang
• 金额: $ 9.85万
• 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: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706507&HistoricalAwards=false
• 关键词: Analytic; Problems; Geometry

9706507 Yang Yang proposes to study three problems in an analytic approach to conformal geometry. The first problem is the variational problem: to minimize the square integral of the Weyl tensor of a four dimensional manifold. The optimal conformal class which achieves the extremal will have a canonical metric which is often Einstein, conformally flat, or half conformally flat. The second problem is to find apriori estimates for solutions of the prescribed scalar curvature problem on spheres of dimension greater than three. It is expected that there will be a new phenomenon to be understood which does not arise in the case for spheres of lower dimensions. The general goal is to find general conditions for the solvability of the Nirenberg problem in these dimensions. The third problem is to investigate regularity property of critical maps of conformally invariant energy functionals. For biharmonic maps in four dimension, such regularity result will permit the construction of biharmonic maps of four dimension which represent an arbitrary homotopy class. Yang has already made good progress on this problem. The project represents an analytic approach to the study of four dimensional spacetime which may or may not model the actual physical spacetime in which we live. The idea is to study various global quantities attached to each spacetime that reflect the local geometric structure. We then look for the spacetime that optimizes these global properties. The ones that do often have nice properties and may allow one to eventually classify the set of all possible spcetimes. The three projects outlined above are technical aspects that need to be developed to carry out such a program.

9706507 杨 杨提出研究三个问题的分析方法，以保形几何。第一个问题是变分问题：最小化四维流形的Weyl张量的平方积分。最优 达到极值的共形类将具有通常是爱因斯坦的、共形平坦的或半共形平坦的正则度量。第二个问题是在大于3维的球面上求出指定数量曲率问题解的先验估计。我们可以预料，将会有一种新的现象需要我们去理解，而这种现象在低维球体的情况下是不会出现的。总的目标是找到 在这些方面的尼伦伯格问题的可解性的一般条件。 第三个问题是研究 共形不变能量泛函的临界映射。对于四维双调和映射，这一正则性结果将允许构造代表任意同伦类的四维双调和映射。 杨在这个问题上已经取得了很好的进展。 该项目代表了一种分析方法来研究四维时空，它可能会或可能不会模拟我们生活的实际物理时空。这个想法是研究各种各样的全球性的量附加到每个时空，反映了当地的几何结构。然后，我们寻找优化这些全局属性的时空。那些做什么 通常具有很好的性质，并且可以允许最终对所有可能的时空的集合进行分类。上文概述的三个项目是需要发展的技术方面，以执行这些项目。 程序.