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 $F\colon \Omega \to \mathbb{R}^m$, where $\Omega \subset \mathbb{R}^n$ is a domain, in $W^{1,n}_{loc}( \Omega, \mathbb{R}^m )$ satisfying $\|DF\|^n = \star F^{*}\omega$ almost everywhere is a restriction of a M\"obius 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 $3 \leq n \leq m$, 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.