A fully nonlinear partial differential equation and its application to the σk-Yamabe problem

The Gursky-Streets equation was introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of the sigma(2)-Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equation with uniform C-1,C-1 estimates for 2k <= n. An important new ingredient is to show the concavity of the operator which holds for all k <= n. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform C-1,C-1 a priori estimates for all the cases n >= 2k. Moreover, we establish the uniqueness of the solution to the degenerate equation for the first time.