Sharp quantitative stability of the M\"obius group among sphere-valued maps in arbitrary dimension

TL;DR

This paper establishes a precise quantitative stability result for the M"obius group among sphere-valued maps in any dimension, extending previous results from three dimensions to higher dimensions with new nonlinear analysis techniques.

Contribution

It provides the first sharp stability estimate for the M"obius group in all dimensions $n \\geq 3$, overcoming nonlinear challenges in higher dimensions.

Findings

Proves a sharp quantitative Liouville theorem for sphere-valued maps.

Extends stability estimates from 3D to higher dimensions $n \\geq 4$.

Introduces new nonlinear analysis techniques using inequalities by Figalli and Zhang.

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 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.