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

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.