Intrinsic potentials in locally harmonic manifolds

TL;DR

This paper introduces the concept of Constant Laplacian potentials (CL-potentials) on spheres and CROSS manifolds, establishing their properties, uniqueness, and relation to energy minimization and point distribution problems.

Contribution

It defines CL-potentials intrinsically on CROSS manifolds, providing a new framework for energy minimization without relying on Euclidean embeddings.

Findings

Existence and uniqueness of CL-potential solutions on spheres.

Integral form of CL-energy and its relation to point configurations.

Asymptotic minimization of Riesz energy configurations for CL-energy.

Abstract

We consider the problem of allocating a finite number of heat sources in the n-dimensional sphere. When only one such source -assumed to be of infinite temperature- is placed and assuming a constant cooling rate in the sphere, we prove that a (essentially) unique solution exists: the Constant Laplacian potential (CL-potential). Actually, this potential can be defined intrinsically in any CROSS (such as the real or complex projective spaces), providing a natural alternative to Riesz's potentials in manifolds lacking a standard isometric embedding into some Euclidean space. We describe an integral form of the corresponding CL-energy for the case of the sphere and prove a relation of minimizing configurations with separation distance and cap discrepancy. It follows that minimal configurations for the Riesz energy are asymptotically minimizing for the CL-energy.