Quasiregular Curves of Small Distortion in Product Manifolds

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 \(\operatorname{vol}_N^\times\in \Omega^n(N)\) 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.