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.

摘要奖项：DMS-0604346主要研究员：李傲冰该奖项支持的项目强调共形不变偏微分方程，包括Yamabe问题的全非线性版本，在规定的共形类中寻求具有恒定标量曲率的黎曼度量，以及边界Yamabe问题的全非线性版本。 其他一些研究方向包括 Trudinger-Wang 不等式在 Hessian 测度上的扩展，以及非局部共形平面的对称曲率函数形式流形的变分性质问题。欧几里得几何提供了直线长度和线对之间角度的测量。线和许多现代几何学都依赖于能够在本地进行这两种测量。 几何学家将坐标的变化描述为“共形”，如果它改变长度但保留角度（想象一下在所有方向上以均匀的量拉伸平面），并且共形变换的研究和应用是几何分析中的一个活跃的研究领域，它依赖于并贡献非线性偏微分方程求解技术的发展。