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/rs, where s>0 and r is Euclidean distance in Rd′. 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 Sd⊂Rd+1, these results are new.