*Accepted Paper*

**Inserted:** 1 jun 2023

**Last Updated:** 13 jul 2024

**Journal:** Trans. Amer. Math. Soc.

**Pages:** 23 pages

**Year:** 2024

**Abstract:**

In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb {S}^{n-1}$ with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch.\ Ration.\ Mech.\ Anal.\ 239(1):219-299, 2021), with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the sharp distance to M\"obius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.