Quasiregular Curves of Small Distortion in Product Manifolds.

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 = N 1 × ⋯ × N k , where each N i is an oriented Riemannian n-manifold, and the calibration vol N × ∈ Ω n (N) is the sum of the Riemannian volume forms 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 ⩽ K ⩽ K 0 and a K-quasiregular vol N × -curve F = (f 1 , ... , f k) : M → N 1 × ⋯ × N k , there exists an index i 0 ∈ { 1 , ... , k } for which the coordinate map f i 0 : M → 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. [ABSTRACT FROM AUTHOR]