Quasiregular Curves of Small Distortion in Product Manifolds

Abstract

We consider, for n 3, K-quasiregular vol × N -curves M → N of small distortion K 1 from oriented Riemannian n-manifolds into Riemannian product manifolds N = N1 × • • • × N k , where each Ni is an oriented Riemannian n-manifold and the calibration vol × N ∈ Ω n (N ) is the sum of the Riemannian volume forms volN i of the factors Ni of N .We show that, in this setting, K-quasiregular curves of small distortion are carried by quasiregular maps. More precisely, there exists K0 = K0(n, k) > 1 having the property that, for 1 K K0 and a K-there exists an index i0 ∈ {1, . . . , k} for which the coordinate map fi 0 : M → Ni 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.