Quasiregular curves

TL;DR

This paper generalizes quasiregular and pseudoholomorphic curves to quasiregular curves between Riemannian manifolds, establishing their fundamental properties, limit behavior, and conditions for discreteness and positivity.

Contribution

It introduces quasiregular curves as a unifying class, proves their quasiminimality, Liouville-type theorem, and stability under limits, extending the theory of quasiregular and holomorphic mappings.

Findings

Quasiregular curves satisfy Gromov's quasiminimality condition.

Bounded quasiregular curves from  to  are constant.

Limit of a sequence of quasiregular curves remains quasiregular.

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.