We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σk-Yamabe problem for locally conformal flat manifolds when k 6= n/2.

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

We give global restrictions on the possible boundaries of compact, orientable, locally conformally flat manifolds of dimension 4k in terms of integrality of eta invariants.

The standard approach to initial data for both analytic and numerical computations of black hole collisions has been to use conformally-flat initial geometry. Among other advantages, this choice allows the simple superposition of holes with arbitrary mass, location and spin. The conformally flat restriction, however, is inappropriate to the study of Kerr holes, for which the standard constant-t...