Discreteness of the minimizers of weakly repulsive interaction energies on Riemannian manifolds

TL;DR

This paper proves that measures minimizing weakly repulsive energies on Riemannian manifolds are supported on discrete sets without concentration points, extending Euclidean results to curved spaces.

Contribution

It extends the understanding of minimizers of weakly repulsive energies from Euclidean spaces to Riemannian manifolds with bounded sectional curvature.

Findings

Minimizers lack concentration points on Riemannian manifolds.

Supports of minimizers are discrete sets.

Extends Euclidean results to curved geometries.

Abstract

It is shown that the supports of measures minimizing weakly repulsive energies on Riemannian manifolds with sectional curvature bounded below do not have concentration points. This extends the results of Bj\"orck and Carrillo, Figalli, and Patacchini for such energies on the Euclidean space, and complements the results about the discreteness of minimizers of the geodesic Riesz energy on the sphere by Bilyk, Dai, and Matzke.