Well-posedness of Hersch-Szeg\H{o}'s center of mass by hyperbolic energy minimization

TL;DR

This paper establishes the existence, uniqueness, and continuous dependence of the hyperbolic center of mass for measures on the unit ball, using a convex energy minimization approach that generalizes classical results like Hersch's lemma.

Contribution

It introduces a new variational characterization of the hyperbolic center of mass as a convex energy minimizer, extending prior results and unifying them under a convexity framework.

Findings

Existence and uniqueness of the hyperbolic center of mass.

Continuous dependence of the center of mass on the measure.

Extension of Hersch's lemma via convexity of a logarithmic kernel.

Abstract

The hyperbolic center of mass of a finite measure on the unit ball with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure. Prior results of this type are extended by characterizing the center of mass as the minimum point of an energy functional that is strictly convex along hyperbolic geodesics. A special case is Hersch's center of mass lemma on the sphere, which follows from convexity of a logarithmic kernel introduced by Douady and Earle.