Local properties of Riesz minimal energy configurations and equilibrium measures

TL;DR

This paper studies the minimal energy configurations of points on smooth manifolds and their separation properties, extending previous results and analyzing equilibrium measures under local smoothness conditions.

Contribution

It establishes optimal separation bounds for Riesz energy minimizers on smooth manifolds and extends these results to greedy configurations and equilibrium measures.

Findings

Optimal separation order for Riesz energy points on manifolds

Extension of Dahlberg's results to broader settings

Separation results for greedy energy points

Abstract

We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $A\subset \mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $s\in [p-2, p-1)$, we prove that the order of separation (as $N\to \infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy' $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.