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 standard S4. Our results show is controlled by L2-norm of Weyl tensor L1-norm Q-curvature, under those quantities are sufficiently small, has positive Yamabe Q-curvature mean-positive. The proof result achieved in two steps. Firstly, we construct quasiconformal map conformally related metrics class. Secondly, apply Ricci flow to establish equivalence from such conformal class