The support of the logarithmic equilibrium measure on sets of revolution in $\R^3$

TL;DR

This paper studies the equilibrium distribution of points minimizing logarithmic energy on surfaces of revolution in 3D, showing they concentrate on the outermost parts of the surface, with analysis reduced to complex plane level lines.

Contribution

It characterizes the support of equilibrium measures on surfaces of revolution, revealing they are supported only on the outermost regions, using a reduction to complex potential theory.

Findings

Limit distributions supported only on outermost surface regions

Support concentrates on positive curvature parts

Reduction to complex plane ellipses level lines

Abstract

For surfaces of revolution $B$ in $\R^3$, we investigate the limit distribution of minimum energy point masses on $B$ that interact according to the logarithmic potential $\log (1/r)$, where $r$ is the Euclidean distance between points. We show that such limit distributions are supported only on the ``out-most'' portion of the surface (e.g., for a torus, only on that portion of the surface with positive curvature). Our analysis proceeds by reducing the problem to the complex plane where a non-singular potential kernel arises whose level lines are ellipses.