A vector equilibrium problem for symmetrically located point charges on a sphere
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 halfline. The two interval case corresponds to a droplet with two disjoint components, and it is analyzed by means of a genus one Riemann surface.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 DOI:
 10.48550/arXiv.2008.01017
 arXiv:
 arXiv:2008.01017
 Bibcode:
 2020arXiv200801017C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematical Physics;
 Mathematics  Complex Variables;
 31A15
 EPrint:
 61 pages, 7 figures