The Gursky-Streets equation and its application to the $\sigma_k$ Yamabe problem

The Gursky-Streets equation are 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 $\sigma_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. 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 $C^{1, 1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformally flat, any solution solution of $\sigma_k$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.