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 $C^{1,1}$ estimates for $2k\le n$. An important new ingredient is to show the concavity of the operator which holds for all $k\le n$. 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 $C^{1,1}$ a priori estimates for all the cases $n\ge 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equation for the first time.

As an application, we prove that if $k\ge 3$ and $M^{2k}$ is conformally flat, any solution of the ${\sigma }_k$-Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.

Gursky-Streets 方程被引入为共形几何中度量结构的测地线方程。这种几何结构在证明唯一性方面发挥了重要作用。${\sigma }_2$- 第 4 维中的 Yamabe 问题。在本文中，我们求解 Gursky-Streets 方程$C^{1,1}$ 估计 $2克\le n$. 一个重要的新成分是显示运算符的凹度，它适用于所有$克\le n$. 我们对凹度的证明在很大程度上依赖于 Garding 的双曲多项式理论和（隔行）多项式的实根理论的结果。与这个凹度一起，我们能够解方程$C^{1,1}$ 所有案例的先验估计$n\ge 2克$. 此外，我们首次建立了退化方程解的唯一性。

作为一个应用，我们证明如果 $克\ge 3$ 和 ${米}^{2克}$ 是共形平坦的，任何解 ${\sigma }_{克}$-Yamabe 问题是对圆形球体的共形微分 ${秒}^{2克}$.