On the Logarithmic Energy of Points on S^2

Abstract

We revisit a classical question: how large is the minimal logarithmic energy of $n$ points on ${\mathbb{S}}^2$ $${\mathcal{E}}_{\log }(n)=\underset{x_1,\dots ,x_n\in {\mathbb{S}}^2}{\min }\sum _{\genfrac{}{}{0ex}{}{i,j=1}{i\ne j}}^n\log \frac{1}{\Vert x_i-x_j\Vert }?$$ Betermin & Sandier (building on work of Sandier & Serfaty) showed that $${\mathcal{E}}_{\log }(n)=\left(\frac{1}{2}-\log 2\right)n^2-\frac{n\log n}{2}+c_{\log }\cdot n+o(n),$$ 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 $c_{\log }\ge -0.223$ to $c_{\log }\ge -0.095$.