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

Abstract

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

This paper has been withdrawn by the authors because the results are now contained (in a reorganized and improved form) in a new paper (arXiv:2101.03846). A mistake in the last part of Lemma 3.2. (ii) (making Theorems 3.5. and 5.4. invalid) has been corrected in the new paper (see Lemma 5.3 and Subsections 5.2 and 5.3 therein)