Lipschitz constants for a hyperbolic type metric under Möbius transformations

2023/09/07 by Wu, Yinping, Wang, Gendi, Jia, Gaili +1
#30C65 #51M10 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2309.03515

Abstract

Let D be a nonempty open set in a metric space (X,d) with ∂ D≠ ∅. Define h_D,c(x,y)=log(1+c(d(x,y))/(√(d_D(x)d_D(y)))), where d_D(x)=d(x,∂ D) is the distance from x to the boundary of D. For every c≥ 2, h_D,c is a metric. In this paper, we study the sharp Lipschitz constants for the metric h_D,c under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

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