Conformally Natural extensions revisited

2011/02/07 by Petersen, Carsten Lunde
#Complex Variables (math.CV) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1102.1470

Abstract

In this note we revisit the notion of conformal barycenter of a measure on \SSⁿ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere \Cbar\isom\SS² to the (hyperbolic) three ball \BB³ and thus to \SS³ by reflection. The construction which was pioneered by Douady and Earle in the case of homeomorphisms actually gives extensions for more general maps such as entire transcendental maps on \C⊂\Cbar. And it works for maps in any dimension.

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