Entire conformal curves with bounded asymptotic energy are affine linear maps, while non-affine curves have unbounded average energy that is strictly increasing for large radii.
Degrees of maps and multiscale geometry
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Entire conformal curves are affine or have super-Euclidean energy growth
Entire conformal curves with bounded asymptotic energy are affine linear maps, while non-affine curves have unbounded average energy that is strictly increasing for large radii.