Fully nonlinear equations and conformal geometry

Jeff Viaclovsky, M.I.T.

Abstract: We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. A special case of this deformation gives a more direct proof of the result of Chang, Gursky and Yang. We give a new conformally invariant condition for positivity of the Paneitz operator, which allows us to give many new examples of manifolds admitting metrics with constant Q-curvature. We will also describe progress on the σ_k-Yamabe problem in dimensions 3 and 4. This is joint work with Matt Gursky.