Conformally Invariant Partial Differential Equations

共形不变偏微分方程

• 批准号: 0604346
• 负责人: Xiuxiong Chen
• 金额: --
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604346&HistoricalAwards=false
• 关键词: Conformally; Invariant; Partial; Differential; Equations

AbstractAward: DMS-0604346Principal Investigator: Aobing LiProjects supported by this award emphasize conformally invariantpartial differential equations, including a fully nonlinearversion of the Yamabe problem that seeks a Riemannian metric withconstant scalar curvature in a prescribed conformal class, and afully nonlinear version of the boundary Yamabe problem. Someother directions of research include an extension of inequalitiesof Trudinger-Wang on the Hessian measure, and a problem on thevariational nature of the symmetric curvature functional formanifolds that are not locally conformally flat.Euclidean geometry provides measurements of both lengths of linesand of angles between pairs of lines, and much of modern geometrydepends upon being able to make both of those measurementslocally. Geometers describe a change of coordinates as"conformal" if it changes lengths but preserves angles (think ofstretching the plane by a uniform amount in all directions), andthe study and application of conformal transformations is anactive area of research in geometric analysis that depends uponand contributes to the development of solution techniques fornonlinear partial differential equations.

AbstractAward: DMS-0604346Principal Investigator: Aobing LiProjects supported by this award emphasize conformally invariantpartial differential equations, including a fully nonlinearversion of the Yamabe problem that seeks a Riemannian metric withconstant scalar curvature in a prescribed conformal class, and afully nonlinear version of the boundary Yamabe problem. 研究的其他一些方向包括在Hessian措施上扩展Trudinger-Wang的不平等，以及关于对称曲率功能的跨性质性质的问题，这些功能函数福利数并非局部在同轴上平坦。渗透性几何形状都可以衡量这些数量的线条和衡量量的几何形式，并衡量了几何和很多几何效率的尺寸。 几何图形将坐标的变化描述为“构型”，如果它改变长度但保持角度（想想在各个方向上以均匀量的平面伸缩平面），而整形态转换的研究和应用是几何学分析中的无性研究领域，则取决于依赖于方案技术的求解技术偏差方程的开发。