Quasiconformal flows on non-conformally flat spheres

We study integral curvature conditions for a Riemannian metric g on S⁴ that quantify the best bilipschitz constant between (S⁴,g) and the standard metric on S⁴. Our results show that the best bilipschitz constant is controlled by the L²-norm of the Weyl tensor and the L¹-norm of the Q-curvature, under the conditions that those quantities are sufficiently small, g has a positive Yamabe constant and the Q-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on S⁴.