Principal Investigator: Aobing Li
Projects supported by this award emphasize conformally invariant partial differential equations, including a fully nonlinear version of the Yamabe problem that seeks a Riemannian metric with constant scalar curvature in a prescribed conformal class, and a fully nonlinear version of the boundary Yamabe problem. Some other directions of research include an extension of inequalities of Trudinger-Wang on the Hessian measure, and a problem on the variational nature of the symmetric curvature functional for manifolds that are not locally conformally flat.
Euclidean geometry provides measurements of both lengths of lines and of angles between pairs of lines, and much of modern geometry depends upon being able to make both of those measurements locally. Geometers describe a change of coordinates as "conformal" if it changes lengths but preserves angles (think of stretching the plane by a uniform amount in all directions), and the study and application of conformal transformations is an active area of research in geometric analysis that depends upon and contributes to the development of solution techniques for nonlinear partial differential equations.
