Fourth Order Equations in Conformal Geometry

共形几何中的四阶方程

• 批准号: 0070526
• 负责人: Paul Yang
• 金额: $ 9.1万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070526&HistoricalAwards=false
• 关键词: Fourth; Order; Equations; Conformal; Geometry

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.

摘要 该项目的主题是发展基于共形几何考虑的四阶方程理论。该项目的第一部分旨在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。第一个结果将是正里奇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可应用我们的四阶方程理论的共形类集。一个具体的问题是在一大类负标量曲率的四维共角结构上找到四阶方程的鞍点解。该项目的第三部分是发展四维双调和映射理论。建议开发特殊共形结构的双调和图的分类。该提案的几个部分有一个共同的新颖特征，即考虑将标量的四阶项与标量本身联系起来的四阶非线性微分方程。该方程来自对四维空间的几何考虑，从拓扑角度来看，这是自然的。该项目将从多个方面对高阶微分方程的发展产生影响。首先，它将引入新技术来研究此类方程的解。其次，它将产生新的方法来求解更传统的几何感兴趣的二阶方程。第三，该结果对四维空间几何有良好的应用。