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Quasiregular Curves of Small Distortion in Product Manifolds (2106.11871v1)

Published 22 Jun 2021 in math.DG, math.AP, math.CV, and math.SG

Abstract: We consider, for $n\ge 3$, $K$-quasiregular $\operatorname{vol}N\times$-curves $M\to N$ of small distortion $K\ge 1$ from oriented Riemannian $n$-manifolds into Riemannian product manifolds $N=N_1\times \cdots \times N_k$, where each $N_i$ is an oriented Riemannian $n$-manifold and the calibration $\operatorname{vol}_N\times\in \Omegan(N)$ is the sum of the Riemannian volume forms $\operatorname{vol}{N_i}$ of the factors $N_i$ of $N$. We show that, in this setting, $K$-quasiregular curves of small distortion are carried by quasiregular maps. More precisely, there exists $K_0=K_0(n,k)>1$ having the property that, for $1\le K\le K_0$ and a $K$-quasiregular $\operatorname{vol}N\times$-curve $F=(f_1,\ldots, f_k) \colon M \to N_1\times \cdots \times N_k$ there exists an index $i_0\in {1,\ldots, k}$ for which the coordinate map $f{i_0}\colon M\to N_{i_0}$ is a quasiregular map. As a corollary, we obtain first examples of decomposable calibrations for which corresponding quasiregular curves of small distortion are discrete and admit a version of Liouville's theorem.

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