The Gursky-Streets equations
arXiv:1707.04689
Abstract
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 -Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of -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 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 geodesic. This in particular gives a straightforward proof of the uniqueness of solutions of -Yamabe problem.