The support of the limit distribution of optimal Riesz energy points on sets of revolution in ${\mathbb{R}}^3$

Abstract

Let A be a compact set in the right-half plane and $\Gamma (A)$ the set in ${\mathbb{R}}^3$ obtained by rotating A about the vertical axis. We investigate the support of the limit distribution of minimal energy point charges on $\Gamma (A)$ that interact according to the Riesz potential 1/r^{s}, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on $\Gamma (A)$ which is supported on the outer boundary of $\Gamma (A)$. We show that there are sets of revolution $\Gamma (A)$ such that the support of the equilibrium measure on $\Gamma (A)$ is {\bf not} the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution $\Gamma (R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0).