Minimal Riesz energy point configurations for rectifiable d -dimensional manifolds

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ R d ′ that interact through the power law (Riesz) potential V = 1 / r s , where s > 0 and r is Euclidean distance in R d ′ . With E s ( A , N ) denoting the minimal energy for such N-point configurat...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2005-05, Vol.193 (1), p.174-204
Main Authors: Hardin, D.P., Saff, E.B.
Format: Article
Language:English
Subjects:
Best-packing
Hausdorff measure
Minimal discrete Riesz energy
Power law potential
Rectifiable manifolds
Uniform distribution of points on a sphere
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Summary:We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ R d ′ that interact through the power law (Riesz) potential V = 1 / r s , where s > 0 and r is Euclidean distance in R d ′ . With E s ( A , N ) denoting the minimal energy for such N-point configurations, we determine the asymptotic behavior (as N → ∞ ) of E s ( A , N ) for each fixed s ⩾ d . Moreover, if A has positive d-dimensional Hausdorff measure, we show that N-point configurations on A that minimize the s-energy are asymptotically uniformly distributed with respect to d-dimensional Hausdorff measure on A when s ⩾ d . Even for the unit sphere S d ⊂ R d + 1 , these results are new.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2004.05.006