Calculus of Variations and Geometric Measure Theory

A. Guerra - X. Lamy - K. Zemas

Optimal Quantitative Stability of the Möbius group of the sphere in all dimensions

created by zemas on 20 Dec 2023
modified on 13 Jan 2024


Submitted Paper

Inserted: 20 dec 2023
Last Updated: 13 jan 2024

Year: 2023


In any dimension $n\geq 3$, we prove an optimal gradient stability estimate for the Möbius group in the conformally invariant critical space $W^{1,n-1}(\mathbb{S}^{n-1};\mathbb{R}^n)$ among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$. The estimate is in terms of 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 $n\geq 4$.