Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 975–990

QUASIREGULAR CURVES

Pekka Pankka

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 68, FI-00014 University of Helsinki, Finland; pekka.pankka 'at' helsinki.fi

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 ≤ m and let M be an oriented Riemannian n-manifold, N a Riemannian m-manifold, and ω ∈ Ωⁿ(N) a smooth closed non-vanishing n-form on N. A continuous Sobolev map f : M → N in W_loc^1,n(M,N) is a K-quasiregular ω-curve for K ≥ 1 if f satisfies the distortion inequality (‖ω‖ o f)‖Df‖ⁿ ≤ K(⋆f^*ω) 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 Rⁿ → R^m are constant. We also prove a limit theorem that a locally uniform limit f : M → N of K-quasiregular ω-curves (f_j : M → N is also a K-quasiregular ω-curve. We also show that a non-constant quasiregular ω-curve f : M → N is discrete and satisfies ⋆f^*ω > 0 almost everywhere, if one of the following additional conditions hold: the form ω is simple or the map f is C¹-smooth.

2010 Mathematics Subject Classification: Primary 30C65; Secondary 32A30, 53C15, 53C57.

Key words: Quasiregular mappings, holomorphic curves, pseudoholomorphic curves and vectors.

Reference to this article: P. Pankka: Quasiregular curves. Ann. Acad. Sci. Fenn. Math. 45 (2020), 975–990.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4534