Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges

TL;DR

This paper characterizes the equilibrium measure support on a sphere with logarithmic interactions in the presence of multiple point charges, linking it to classical quadrature domains via stereographic projection.

Contribution

It provides a novel description of equilibrium measures on the sphere with external point charges, connecting potential theory with classical quadrature domains.

Findings

Support complement is a union of quadrature domains.

Supports are determined by external point charges.

Stereographic projection relates sphere problem to plane quadrature domains.

Abstract

With the sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges.