LAUSR

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 $$\sigma _2$$ σ 2 -Yamabe problem. The geodesic equation of Gursky–Streets’ metric is a fully nonlinear degenerate elliptic equation. Using this geometric structure and the geodesic equation, Gursky–Streets proved an important result in conformal geometry, that the solution of the $$\sigma _2$$ σ 2 -Yamabe problem is unique (the existence of such a solution was known more than a decade ago). A key ingredient is the convexity of Chang–Yang’s $${{\mathcal {F}}}$$ F -functional along the (smooth) geodesic. However Gursky–Streets have only proved uniform $$C^{0, 1}$$ C 0 , 1 regularity for a perturbed equation and it turns out that the uniform $$C^{1, 1}$$ C 1 , 1 regularity is very delicate. Without such a uniform $$C^{1, 1}$$ C 1 , 1 regularity, Gursky–Streets arguments of the uniqueness theorem are very complicated. In this paper we establish the uniform $$C^{1, 1}$$ C 1 , 1 regularity of the Gursky–Streets equation. In the course of deriving regularity, we also obtain very interesting and new convexity regarding matrices in $$\Gamma _2^+$$ Γ 2 + . As an application, we can establish strictly the convexity of $${{\mathcal {F}}}$$ F -functional along $$C^{1, 1}$$ C 1 , 1 geodesic. This gives a straightforward proof of the uniqueness of solutions of $$\sigma _2$$ σ 2 -Yamabe problem.