Quasiconformal Flows on non-Conformally Flat Spheres

Abstract

We study integral curvature conditions for a Riemannian metric g on S 4 that quantify the best bilipschitz constant between (S 4 , g) and the standard metric on S 4 . Our results show that the best bilipschitz constant is controlled by the L 2 -norm of the Weyl tensor and the L 1 -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 S 4 .

Some aspects of Ricci flow on the 4-sphere

“…In particular, we study the decay of the curvature quantities |W |, |E|, and |R − R|, or equivalently the decay of the reduced curvature tensor Riem − 1 24 Rg ⊙ g , in the terminology of the abstract. We also remark that the monotonicity result of Section 3 has been applied in some recent work of Chang-Prywes-Yang [CPY21]to establish the bi-Lipschitz equivalence of a class of metrics to the canonical metric on S 4 under some suitable curvature conditions.…”