Analytic Problems in Geometry

杨洋提出用解析方法研究共形几何中的三个问题。第一个问题是变分问题：最小化一个四维流形的Weyl张量的平方积分。达到极值的最优共形类将有一个标准度规，通常是爱因斯坦，共形平坦，或半共形平坦。第二个问题是在大于3维的球面上寻找给定标量曲率问题解的先验估计。预计将会有一种新的现象被理解，而这种现象在较低维度的球体中不会出现。总的目标是在这些维度中找到尼伦堡问题可解的一般条件。第三个问题是研究共形不变能量泛函的临界映射的正则性。对于四维双调和映射，这种正则性结果将允许构造表示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，它可能会也可能不会模拟我们生活的实际物理时空。这个想法是研究附着在每个时空上的反映局部几何结构的各种全局量。然后我们寻找优化这些全局属性的时空。这样做的那些通常具有很好的属性，并且可能允许人们最终对所有可能的空间时间集合进行分类。上述三个项目是执行这一方案需要发展的技术方面。

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.