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Let A be a compact point set in the right half of the xy plane and Γ(A) the set in R3 obtained by rotating A about the y axis. We investigate the support of the limit distribution of minimal energy point charges on Γ(A) that interact according to the Riesz potential 1/rs, 0

Let A be a compact point set in the right half of the xy plane and Γ(A) the set in R3 obtained by rotating A about the y axis. We investigate the support of the limit distribution of minimal energy point charges on Γ(A) that interact according to the Riesz potential 1/rs, 0

For sets of revolution B in R3, 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 “outmost” 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 nonsingular potential kernel arises, the level lines of which are ellipses. [ABSTRACT FROM AUTHOR]

For sets of revolution B in R3, 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 “outmost” 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 nonsingular potential kernel arises, the level lines of which are ellipses. [ABSTRACT FROM AUTHOR]

Configurations of N points on the two-sphere that are stable with respect to the Riesz s-energy have a structure that is largely hexagonal. These stable configurations differ from the configurations with the lowest reported N-point s-energy in the location and structure of defects within this hexagonal structure. These differences in energy between the stable and minimal configuration suggest that energy scale at which defects play a role. This work uses numerical experiments to report this difference as a function of N, allowing us to infer the energy scale at which defects play a role. This work is presented in the context of established estimates for the minimal N-point energy, and in particular we identify terms in these estimates that likely reflect defect structure. [ABSTRACT FROM AUTHOR]