Spherical Arc-Length as a Global Conformal Parameter for Analytic Curves in the Riemann Sphere

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Date
2016-11-11MFO Scientific Program
Research in Pairs 2016Series
Oberwolfach Preprints;2016,21Author
Gauthier, Paul Montpetit
Nestoridis, Vassili
Papadopoulos, Athanase
Metadata
Show full item recordOWP-2016-21
Abstract
We prove that for every analytic curve in the complex plane $\mathbb{C}$, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in $\mathbb{R}^n$ and $\mathbb{C}^n$ and we discuss the situation of curves in the Riemann sphere $\mathbb{C} \cup \{\infty\}.$