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

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium measure is the uniform distribution on a sphere. We develop general necessary and 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 a certain dimension reduction phenomenon occurs: the support of the equilibrium measure becomes a sphere. We also briefly discuss the relationship between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk–Hecke formula, and the calculus of hypergeometric functions.