Fourth Order Equations in Conformal Geometry

该项目的主要主题是发展由共形几何考虑产生的四阶方程的理论。该项目的第一部分试图将四阶方程应用于共形几何中更自然但更困难的完全非线性偏微分方程组的正则化方程。第一个结果将是存在正Ricci曲率的共形度量的自然共形准则。该项目的第二部分试图扩大四阶方程的小时理论可以应用到的保形类的集合。一个具体的问题将是在一大类负标量曲率的四维共形结构上找到四阶方程的鞍点解。项目的第三部分是发展四维双调和映射理论。提出了一种特殊构象的双调和映射的分类方法。该方案的几个部分有一个共同的新特点，即考虑一个四阶非线性微分方程，该方程将标量的四阶项与标量本身联系起来，该方程来自于从拓扑角度看是自然的四维空间的几何考虑。该项目将在几个方面对高阶微分方程组的发展产生影响。首先，它将引入新的技术来研究这类方程的解。其次，它将为求解更传统的二阶几何方程提供新的方法。第三，这些结果已经应用于四维空间的几何。

ABSTRACTThe main theme of the project is to develop the theory of fourth orderequations arising from conformal geometric considerations. The firstpart of the project seeks to apply the fourth order equation as regularizing equations of the more natural but more difficult fullynonlinear partial differential equations in conformal geometry.The first consequence will be natural conformal criteria for theexistence of a conformal metric of positive Ricci curvature. The secondpart of the project seeks to widen the set of conformal classes to whichour theory of fourth order equations may be applied. A concrete problemwould be to find saddle point solutions of the fourth order equationson a large class of four dimensional conformal structures of negativescalar curvature. The third part of the proposed project is to developthe theory of biharmonic maps in four dimension. It is proposed todevelop a classification of biharmonic maps of special conformalstructures. The several parts of this proposal share a common novel feature toconsider a fourth order nonlinear differential equation that relatesfourth order term of a scalar quantity to the scalar quantity itself.The equation came from geometric considerations for a four dimensionalspace that are natural from topological perspective. The projectwill have an impact on the development of higher order differentialequations in several ways. First it will introduce new techniquesto study the solutions of such equations. Secondly, it will yield newmethods to solve the more traditional second order equations of geometricinterest. Thirdly, the results have aleady yield application to thegeometry of four dimensional space.