Discrete and Continuous Green Energy on Compact Manifolds

TL;DR

This paper investigates how the Green function for the Laplacian on compact Riemannian manifolds can be used to generate well-distributed points, showing that energy minimizers tend to distribute uniformly, especially on locally harmonic manifolds.

Contribution

It establishes the asymptotic uniform distribution of Green energy minimizers on compact Riemannian manifolds, with a focus on locally harmonic cases, advancing understanding of point distribution methods.

Findings

Green energy minimizers are asymptotically uniformly distributed.

The results apply notably to locally harmonic manifolds.

Provides a new approach to point distribution on manifolds.

Abstract

In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally harmonic manifolds.