A rescaling principle for quasiregular curves with applications to hyperbolicity

Abstract

We prove a Miniowitz--Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder--Iliashenko--Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi--Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.