A vector equilibrium problem for symmetrically located points on the sphere

I will discuss an equilibrium problem with a finite number of fixed point charges on the unit sphere. The fixed charges repel a large number of small free charges that in the large $n$ limit will fill out a two dimensional part of the sphere called the droplet. The motherbody of the droplet is a measure that is supported on a onedimensional subset generating the same electrostatic potential in the exterior of the droplet. After projecting onto the complex plane the support of the motherbody is a system of contours and we conjecture that it is characterized by a minmax property for a vector equilibrium problem. As a first step in this direction we consider a symmetric situation where the fixed points are in a symmetric position around the north pole. The motherbody is supported on a number of meridians that project onto half-rays in the complex plane. We formulate and study the vector equilibrium problem in this situation. Along the way we find connections with iterated balayage, Riemann surfaces, and Muttalib Borodin ensembles. There are probably interesting multiple orthogonal polynomials as well. This is joint work with Juan Criado del Rey (Leuven).