Conformally Invariant Partial Differential Equations

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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.