Stability estimates for the conformal group of $\mathbb{S}^{n-1}$ in dimension $n\geq 3$

TL;DR

This paper establishes a quantitative stability estimate for Möbius transformations on the sphere in dimensions three and higher, linking geometric rigidity with conformal-isoperimetric deficits.

Contribution

It provides a local stability estimate for conformal maps on spheres, connecting deviations from Möbius transformations to conformal deficits in a quantitative manner.

Findings

Optimal stability estimates are derived.

The results relate to geometric rigidity of the special orthogonal group.

The stability estimate is of local nature, involving average conformal deficits.

Abstract

The purpose of this paper is to exhibit a quantitative stability result for the class of M\"obius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The main estimate is of local nature and asserts that for a Lipschitz map that is apriori close to a M\"obius transformation, an average conformal-isoperimetric type of deficit controls the deviation (in an average sense) of the map in question from a particular M\"obius map. The optimality of the result together with its link with the geometric rigidity of the special orthogonal group are also discussed.