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A Vector Equilibrium Problem for Symmetrically Located Point Charges on a Sphere

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Abstract

We study the equilibrium measure on the two-dimensional sphere in the presence of an external field generated by \(r+1\) equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or the full half-line. The two interval case corresponds to a droplet with two disjoint components, and it is analyzed by means of a genus one Riemann surface.

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Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, 3010, Leuven, Belgium

    Juan G. Criado del Rey & Arno B. J. Kuijlaars

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  1. Juan G. Criado del Rey
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  2. Arno B. J. Kuijlaars
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Correspondence to Arno B. J. Kuijlaars.

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Communicated by Edward B. Saff.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Juan G. Criado del Rey: Supported by FWO Flanders Project EOS 30889451.

Arno B. J. Kuijlaars: Supported by long term structural funding-Methusalem Grant of the Flemish Government, and by FWO Flanders Projects EOS 30889451, G.0864.16 and G.0910.20.

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Criado del Rey, J.G., Kuijlaars, A.B.J. A Vector Equilibrium Problem for Symmetrically Located Point Charges on a Sphere. Constr Approx 55, 775–827 (2022). https://doi.org/10.1007/s00365-022-09566-5

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  • DOI: https://doi.org/10.1007/s00365-022-09566-5

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