*Published Paper*

**Inserted:** 17 sep 2022

**Last Updated:** 17 sep 2022

**Journal:** Inventiones mathematicae

**Volume:** 230

**Number:** Issue 1

**Pages:** 375-461

**Year:** 2022

**Doi:** 10.1007/s00222-022-01128-7

**Abstract:**

We investigate both linear and nonlinear stability aspects of rigid motions (resp. MÃ¶bius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric (resp. conformal) deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. The focus is mostly on the case $n=3$ (where it is explained why the estimates are optimal in their corresponding settings), but we also address the necessary adaptations for the results in higher dimensions. We also obtain linear stability estimates for both cases in all dimensions. These can be regarded as Korn-type inequalities for the combination of the quadratic form associated with the isometric (resp. conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.