Fourth Order Equations in Conformal Geometry

本项目的主题是发展由共形几何考虑引起的四阶方程理论。第一部分是应用四阶方程作为共形几何中更自然但更困难的完全非线性偏微分方程的正则化方程，第一个结果是正Ricci曲率的共形度量存在的自然共形准则.该项目的第二部分旨在扩大一套共形类whichour理论的四阶方程可能会被应用。一个具体的问题是在一大类负量曲率的四维共形结构上求四阶方程的鞍点解。第三部分是发展四维双调和映射的理论。本文提出了一种具有特殊共形结构的双调和映射的分类方法。这几个部分有一个共同的特点，即考虑一个四阶非线性微分方程，它把标量的四阶项与标量本身联系起来，这个方程来自于从拓扑角度看是自然的四维空间的几何考虑.该项目在几个方面对高阶微分方程的发展产生了影响。首先介绍研究这类方程解的新方法。其次，它将产生新的方法来解决更传统的二阶几何方程。第三，这些结果在四维空间几何中已有应用。

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.