Liouville's theorem in calibrated geometries

Abstract

We consider the following extension of the classical Liouville theorem: A calibration $\omega \in {\Lambda }^n{\mathbb{R}}^m$, where $3\le n\le m$, has the Liouville property if a Sobolev mapping $F:\Omega \to {\mathbb{R}}^m$, where $\Omega \subset {\mathbb{R}}^n$ is a domain, in $W_{loc}^{1,n}(\Omega ,{\mathbb{R}}^m)$ satisfying $\Vert DF{\Vert }^n=\star F^{\ast }\omega$ almost everywhere is a restriction of a Möbius transformation ${\mathbb{S}}^m\to {\mathbb{S}}^m$. We show that, for $m\ge 5$, every calibration in ${\Lambda }^{m-2}{\mathbb{R}}^m$ has the Liouville property and, in low dimensions, a calibration $\omega \in {\Lambda }^n{\mathbb{R}}^m$ has the Liouville property for $3\le n\le m\le 6$ unless $\omega$ is face equivalent to the Special Lagrangian. In these cases, the Liouville property stems from isoperimetric rigidity of these mappings together with a classification of calibrations whose conformally flat calibrated submanifolds are flat. We also show that, for $3\le n\le m$, the calibrations with the Liouville property form a dense $G_{\delta }$ set in the space of calibrations. As an application, we consider factorization of more general quasiregular curves and stability of quasiregular curves of small distortion.

43 pages