Conformally Invariant Partial Differential Equations

摘要奖：DMS-0604346首席研究员：该奖项支持的项目强调共形不变的偏微分方程，包括寻求在规定的共形类中具有常标量曲率的黎曼度量的Yamabe问题的完全非线性版本，以及完全非线性版本的边界Yamabe问题。其他一些研究方向包括推广Trudinger-Wang关于Hessian测度的不等式，以及关于非局部共形平坦的对称曲率泛函流形的变分性质的问题。欧几里德几何提供了直线长度和直线对之间夹角的测量，而许多现代几何依赖于能够在局部进行这两种测量。几何学家将改变长度但保持角度的坐标变化描述为“共形”(考虑在所有方向上均匀地拉伸平面)，而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，它依赖于并有助于非线性偏微分方程解技术的发展。

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.