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 nm 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 : MN in Wloc1,n(M,N) is a K-quasiregular ω-curve for K ≥ 1 if f satisfies the distortion inequality (‖ω‖ o f)‖DfnK(⋆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 RnRm are constant. We also prove a limit theorem that a locally uniform limit f : MN of K-quasiregular ω-curves (fj : MN is also a K-quasiregular ω-curve. We also show that a non-constant quasiregular ω-curve f : MN 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

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