The Support of the Logarithmic Equilibrium Measure on Sets of Revolution in R

For surfaces of revolution B in R, 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.