MINIMAL RIESZ ENERGY POINT CONFIGURATIONS FOR RECTIFIABLE d-DIMENSIONAL MANIFOLDS

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each fixed s ≥ d. Moreover, if A has positive d-dimensional Hausdorff measure, we show that N -point configurations on A that minimize the s-energy are asymptotically uniformly distributed with respect to d-dimensional Hausdorff measure on A when s ≥ d. Even for the unit sphere S ⊂ R, these results are new.