Optimal quantitative stability of the Möbius group of the sphere in all dimensions

Abstract

In any dimension $n\ge 3$, we prove an optimal stability estimate for the Möbius group among maps $u:{\mathbb{S}}^{n-1}\to {\mathbb{R}}^n$, of the form ${\inf }_{\lambda >0,\varphi \in \mathrm{M}\ddot{\mathrm{o}}\mathrm{b}({\mathbb{S}}^{n-1})}{\int }_{{\mathbb{S}}^{n-1}}{\left|\frac{1}{\lambda }{\nabla }_{T}u-{\nabla }_T\varphi \right|}^{n-1}d{\mathcal{H}}^{n-1}\le C_n{\mathcal{E}}_{n-1}(u).$ Here, ${\mathcal{E}}_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u({\mathbb{S}}^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the Möbius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\ge 4$.

46 pages

References in corpus (3)

• Regularizing properties of n-Laplace systems with antisymmetric potentials in Lorentz spaces
• Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree
• Sharp quantitative stability of the Möbius group among sphere-valued maps in arbitrary dimension