Quasiconformal Flows on non-Conformally Flat Spheres

TL;DR

This paper establishes conditions under which a Riemannian metric on the 4-sphere is bilipschitz close to the standard metric, using integral curvature bounds and Ricci flow techniques.

Contribution

It introduces new integral curvature conditions involving the Weyl tensor and Q-curvature that control the bilipschitz constant to the standard sphere.

Findings

Bilipschitz constant is bounded by curvature norms.

Constructs quasiconformal maps in positive Yamabe class.

Uses Ricci flow to relate conformal classes.

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 $S^4$.