Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|^−s with d−2?s<d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s=d−2 yields interesting phenomena, which we investigate in detail. A weak^∗ asymptotic analysis is provided as s→⁺(d−2).