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

TL;DR

This paper establishes optimal rigidity estimates for isometric and conformal maps from spheres to Euclidean space, incorporating both geometric and distortion deficits, with results particularly sharp in three dimensions.

Contribution

It introduces combined stability estimates involving isometric/conformal deficits and isoperimetric deficits for maps from spheres, extending and optimizing previous rigidity results.

Findings

Optimal estimates for $n=3$

Linear stability in all dimensions

Extension to higher Sobolev norms for $n eq 3$

Abstract

We investigate both linear and nonlinear stability aspects of rigid motions (resp. M\"obius 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 $n\geq 4$ 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.