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

D. P. Hardin·E. B. Saff·H. Stahl
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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.

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D. P. Hardin, E. B. Saff, H. Stahl. 2006-05-09. The support of the logarithmic equilibrium measure on sets of revolution in $\R^3$. https://doi.org/10.1063/1.2435084

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