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.
All Science Journal Classification (ASJC) codes
- Conformal geometry
- Quasiconformal maps