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The Gursky-Streets equation are introduced as the geodesic equation of ametric structure in conformal geometry. This geometric structure has played asubstantial role in the proof of uniqueness of $\sigma_2$ Yamabe problem indimension four. In this paper we solve the Gursky-Streets equations withuniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is toshow the concavity of the operator which holds for all $k\leq n$. Our proof ofthe concavity heavily relies on Garding's theory of hyperbolic polynomials andresults from the theory of real roots for (interlacing) polynomials. Togetherwith 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, weestablish the uniqueness of the solution to the degenerate equations for thefirst time.  As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformallyflat, any solution solution of $\sigma_k$ Yamabe problem is conformaldiffeomorphic to the round sphere $S^{2k}$.

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