The Gursky-Streets equations

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $\sigma_2$-Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform $C^{0, 1}$ regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of $\sigma_2$-Yamabe problem is unique. A key ingredient is the convexity of Chang-Yang's $\cF$-functional along the (smooth) geodesic, in view of Gursky-Streets metric and a weighted Poincare inequality of B. Andrews on manifolds with positive Ricci curvature. In this paper we establish uniform $C^{1, 1}$ regularity of the Gursky-Streets' equation. As an application, we can establish strictly the geometric structure in terms of Gursky-Streets' metric, in particular the convexity of $\cF$-functional along $C^{1, 1}$ geodesic. This in particular gives a straightforward proof of the uniqueness of solutions of $\sigma_2$-Yamabe problem.