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-0604346首席研究员：李傲兵该奖项支持的项目强调共形不变偏微分方程，包括Yamabe问题的完全非线性版本，该问题在规定的共形类中寻求具有常数标量曲率的黎曼度量，以及边界Yamabe问题的完全非线性版本。 其他一些方向的研究包括一个扩展的不等式Trudinger-王的海森措施，和一个问题的变分性质的对称曲率功能形式流形是不是局部共形平坦。欧几里德几何提供的测量长度的线和线对之间的角度，和现代geometrydepends on being able to make both of those measures locally. 几何学将坐标的变化描述为“保形”，如果它改变了长度但保持了角度（想想在所有方向上以均匀的量拉伸平面），保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，它依赖于并有助于非线性偏微分方程求解技术的发展。