Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?

TL;DR

This paper investigates conditions under which Riesz energy equilibria with radial external fields are supported on spheres, providing complete characterizations and linking to constrained optimization problems.

Contribution

It establishes necessary and sufficient conditions for spherical support of equilibrium measures and characterizes when dimension reduction occurs for power-law fields.

Findings

Equilibrium support is a sphere under specific external field conditions.

Complete characterization of power values leading to dimension reduction.

Connections made between Riesz energy problems and constrained optimization.

Abstract

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.