*Submitted Paper*

**Inserted:** 20 dec 2023

**Last Updated:** 13 jan 2024

**Year:** 2023

**Abstract:**

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 $n\geq 4$.

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