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

Abstract

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 σ2-Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equation with uniform C1,1 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 C1,1a priori estimates for all the cases n≥2k. Moreover, we establish the uniqueness of the solution to the degenerate equation for the first time.As an application, we prove that if k≥3 and M2k is conformally flat, any solution of the σk-Yamabe problem is conformal diffeomorphic to the round sphere S2k.