Subharmonic Kernels and Energy Minimizing Measures, with Applications to the Flat Torus

TL;DR

This paper investigates energy minimization problems involving subharmonic kernels on compact metric spaces, extending classical electrostatics, and applies results to the flat torus and group-invariant kernels, revealing conditions for minimal measures and positive definiteness.

Contribution

It introduces a framework for analyzing energy minimization with subharmonic kernels, extending classical results, and applies these to homogeneous manifolds and the flat torus, establishing new conditions for minimal measures and positive definiteness.

Findings

Minimizing measures have constant potential outside a small set.

The Riesz kernel is minimized by the uniform measure on the flat torus for certain parameters.

Positive definiteness of kernels is linked to the nonnegativity of Fourier coefficients.

Abstract

We study the minimization of the energy integral $I_K(\mu) = \int_{\Omega} \int_{\Omega} K(x,y) d\mu(x) d\mu(y)$ over all Borel probability measures $\mu$, where $(\Omega,\rho)$ is a compact connected metric space and $K:\Omega^2 \to [0,\infty]$ is continuous in the extended sense. We focus on kernels $K$ which are subharmonic, which we define so that the potential $U_K^\mu(x) = \int_{\Omega} K(x,y) d\mu(y)$ satisfies a maximum principle on $\Omega\setminus{\rm supp}{\mu}$. This extends the classical electrostatics minimization problem for logarithmic energy $\int_{\Omega}\int_{\Omega}\log\left(\frac{1}{||x-y||}\right)$, which is used heavily as a tool in approximation theory. Using properties of minimizing measures, we show that if the singularities of the subharmonic kernel $K$ are such that $K$ is regular, then $K$ is positive definite, and $\mu$ is a minimizing measure if and only if its potential is constant (outside of a small exceptional set).We then apply this result to group invariant kernels on compact homogeneous manifolds. In this case, the uniform measure $\sigma$ has constant potential, so subharmonicity implies that this is the minimizing measure. Finally, we look at the case of the $d$-dimensional flat torus $T^d$. We use our results to see that the Riesz kernel $K_s(x,y) = {\rm sign}(s)\rho(x,y)^{-s}$ is minimized by $\sigma$ (and thus positive definite) when $d > s \geq d-2$. Additionally, the positive definiteness gives us a condition which implies that the multivariate Fourier series of a function $f:[0,\pi]^d \to [0,\infty]$ has nonnegative coefficients.