A vector equilibrium problem for symmetrically located point charges on a sphere

TL;DR

This paper investigates the equilibrium measure on a sphere influenced by symmetrically placed point charges, revealing phase transitions in the droplet's support characterized via a vector equilibrium problem and complex analysis techniques.

Contribution

It introduces a vector equilibrium problem framework to analyze the droplet support and describes phase transitions in the equilibrium measure on the sphere.

Findings

Support of the equilibrium measure can be a finite interval, two intervals, or the full half-line.

The two-interval case is analyzed using genus one Riemann surface techniques.

The model exhibits two phase transitions in the droplet's structure.

Abstract

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by r+1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or the full half-line. The two interval case corresponds to a droplet with two disjoint components, and it is analyzed by means of a genus one Riemann surface.