Quasiregular Curves of Small Distortion in Product Manifolds

Abstract

We consider, for $$n\geqslant 3$$ , K-quasiregular $$\mathrm {vol}_N^\times $$ -curves $$M\rightarrow N$$ of small distortion $$K\geqslant 1$$ 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 $$\mathrm {vol}_N^\times \in \Omega ^n(N)$$ is the sum of the Riemannian volume forms $$\mathrm {vol}_{N_i}$$ 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 $$1\leqslant K\leqslant K_0$$ and a K-quasiregular $$\mathrm {vol}_N^\times $$ -curve $$F=(f_1,\ldots , f_k) :M \rightarrow N_1\times \cdots \times N_k$$ , there exists an index $$i_0\in \{1,\ldots , k\}$$ for which the coordinate map $$f_{i_0}:M\rightarrow N_{i_0}$$ 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.