Quasiregular curves

Abstract

We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let $n\le m$ and let $M$ be an oriented Riemannian $n$-manifold, $N$ a Riemannian $m$-manifold, and $\omega \in \Omega^n(N)$ a smooth closed non-vanishing $n$-form on $N$. A continuous Sobolev map $f\colon M \to N$ in $W^{1,n}_{\mathrm{loc}}(M,N)$ is a $K$-quasiregular $\omega$-curve for $K\ge 1$ if $f$ satisfies the distortion inequality $(\lVert\omega\rVert\circ f)\lVert Df\rVert^n \le K (\star f^* \omega)$ almost everywhere in $M$. We prove that quasiregular curves satisfy Gromov's quasiminimality condition and a version of Liouville's theorem stating that bounded quasiregular curves $\mathbb R^n \to \mathbb R^m$ are constant. We also prove a limit theorem that a locally uniform limit $f\colon M \to N$ of $K$-quasiregular $\omega$-curves $(f_j \colon M\to N)$ is also a $K$-quasiregular $\omega$-curve. We also show that a non-constant quasiregular $\omega$-curve $f\colon M \to N$ is discrete and satisfies $\star f^*\omega >0$ almost everywhere, if one of the following additional conditions hold: the form $\omega$ is simple or the map $f$ is $C^1$-smooth.