Möbius parametrizations of curves in $${\mathbb{R}}^n$$

Abstract

We use Ahlfors’ definition of Schwarzian derivative for curves in euclidean spaces to present new results about Mobius or projective parametrizations. The class of such parametrizations is invariant under compositions with Mobius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation \(u^{\prime\prime} + \frac{1}{4}k^{2}u = 0\), where k = k(s) is the curvature as a function of arclength.