2019/09/18 by Pekka Pankka, Pankka, Pekka · 6 citations
Mathematics · Engineering · #Algebraic Geometry and Number Theory #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.1909.08221
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\colon M → N in W1,nloc(M,N) is a K-quasiregular ω-curve for K≥ 1 if f satisfies the distortion inequality (‖ω‖∘ 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 \mathbb Rn → \mathbb Rm are constant. We also prove a limit theorem that a locally uniform limit f\colon M → N of K-quasiregular ω-curves (fj \colon M→ N) is also a K-quasiregular ω-curve. We also show that a non-constant quasiregular ω-curve f\colon 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.