Entire conformal curves are affine or have super-Euclidean energy growth

Abstract

We prove that entire conformal curves $\mathbb{R}^n \rightarrow \mathbb{R}^m$ fall into two classes: either the curve is affine or the average energy in a ball is strictly increasing for large radii and diverges to infinity. This rigidity follows from a blow-down argument and the strong interaction of the generalized Cauchy--Riemann equations with calibrated geometries and the sharp isoperimetric inequality for integral currents due to Almgren. As an application, we prove that every entire Lipschitz conformal curve is affine. This can be considered a higher-dimensional analog of the statement that an entire holomorphic function with a bounded complex differential is affine. We also recall that a certain punctured cone over a Legendrian torus provides a submanifold in $\mathbb{R}^n$, for $n \geq 3$, for which an entire conformal curve yields a conformal covering map. As a related result, we prove that if a non-constant entire conformal curve factors through an $n$-dimensional submanifold, then the submanifold is calibrated and conformally equivalent to a flat non-compact quotient of $\mathbb{R}^n$.