Abstract. We investigate the energy of arrangements of N points on the surface of a sphere in R 3 , interacting through a power law potential V = r α , −2 < α < 2, where r is Euclidean distance. For α = 0, we take V = log(1/r). An area-regular partitioning scheme of the sphere is devised for the purpose of obtaining bounds for the extremal (equilibrium) energy for such points. For α = 0, finer estimates are obtained for the dominant terms in the minimal energy by considering stereographical projections on the plane and analyzing certain logarithmic potentials. A general conjecture on the asymptotic form (as N → ∞) of the extremal energy, along with its supporting numerical evidence, is presented. Also we introduce explicit sets of points, called "generalized spiral points", that yield good estimates for the extremal energy. At least for N ≤ 12, 000 these points provide a reasonable solution to a problem of M. Shub and S. Smale arising in complexity theory.

No abstract

No abstract

No abstract

A revision of the last appendix of the paper "Existence and Regularity for an Energy Maximization Problem in Two Dimensions" by S.Kamvissis and E.