On the Logarithmic Energy of Points on S^2

DOI10.1007/S11854-022-0225-4arXiv2011.04630MaRDI QIDQ6353347

Stefan Steinerberger

Publication date: 9 November 2020

Abstract: We revisit a classical question: how large is the minimal logarithmic energy of $n$ points on $mathbb{S}^2$ ${\displaystyle }$ mathcal{E}_{log}(n) = min_{x_1, dots, x_n in mathbb{S}^2} quad sum_{i,j =1 atop i eq j}^{n}{ log{frac{1}{|x_i-x_j|}} } ? ${\displaystyle }$ Betermin & Sandier (building on work of Sandier & Serfaty) showed that ${\displaystyle }$mathcal{E}_{log}(n) = left( frac{1}{2} - log{2} ight)n^2 - frac{n log{n}}{2} + c_{log} cdot n + o(n),${\displaystyle }$ where the constant $c_{log}$ is characterized by a certain renormalized minimization problem. Brauchart, Hardin & Saff conjectured a closed form expression for $c_{log}$ ($sim-0.05$) assuming analytic continuation. We describe a simple renormalization approach that results in a purely local problem involving superpositions of Gaussians. In particular, if the hexagonal lattice minimizes Gaussians energy, this would prove that $c_{log}$ indeed coincides with the conjectured value. We also improve the lower bound from $c_{log}geq-0.223$ to $c_{log}geq-0.095$.

Mathematics Subject Classification ID

Potentials and capacities on other spaces (31C15)

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