Minimum Energy Problem on the Hypersphere

TL;DR

This paper solves the minimum energy problem on a sphere with external fields by explicitly determining the extremal measure's support and density, with applications to point charges and quadratic external fields.

Contribution

It provides explicit solutions for the extremal measure's support and density on the sphere under external fields, advancing potential theory methods.

Findings

Explicit support of extremal measure derived

Density of extremal measure obtained in closed form

Applications to point charge and quadratic external fields

Abstract

We consider the minimum energy problem on the unit sphere $\mathbb S^{d-1}$ in the Euclidean space $\mathbb R^d$, $d\geq 3$, in the presence of an external field $Q$, where the charges are assumed to interact according to Newtonian potential $1/r^{d-2}$, with $r$ denoting the Euclidean distance. We solve the problem by finding the support of the extremal measure, and obtaining an explicit expression for the density of the extremal measure. We then apply our results to an external field generated by a point charge of positive magnitude, placed at the North Pole of the sphere, and to a quadratic external field.