Rigidity estimates for isometric and conformal maps from ${\mathbb{S}}^{n-1}$ to ${\mathbb{R}}^n$

Abstract

We investigate both linear and nonlinear stability aspects of rigid motions (resp. Möbius 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. We mostly focus on the case $n=3$, where we also explain why the estimates are optimal in their corresponding settings. In the isometric case the estimate holds true also when $n=2$ and generalizes in dimensions $n\ge 4$ as well, if one requires apriori boundedness in a certain higher Sobolev norm. 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.

61 pages, final version after reviewing process. Accepted in Inventiones Mathematicae

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