
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 and let be an oriented Riemannian -manifold, a Riemannian -manifold, and a smooth closed non-vanishing -form on . A continuous Sobolev map in is a -quasiregular -curve for if satisfies the distortion inequality almost everywhere in . We prove that quasiregular curves satisfy Gromov's quasiminimality condition and a version of Liouville's theorem stating that bounded quasiregular curves are constant. We…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Numerical Analysis Techniques
