An equilibrium problem on the sphere with two equal charges

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

arxiv: 1907.04801 · v1 · pith:BJOT255Znew · submitted 2019-07-10 · 🧮 math.CV

An equilibrium problem on the sphere with two equal charges

Juan G. Criado del Rey , Arno B. J. Kuijlaars This is my paper

Pith reviewed 2026-05-24 23:14 UTC · model grok-4.3

classification 🧮 math.CV

keywords equilibrium measuredropletspherepoint chargesstereographic projectionellipsemother body

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The pith

For sufficiently large equal charges the equilibrium droplet on the sphere becomes simply connected with boundary that stereographic projection maps to an ellipse.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the equilibrium measure on the sphere whose external field is produced by two equal point charges. Earlier results had established that small charges leave the complement of the support consisting of two spherical caps. For larger charges the support, called the droplet, becomes simply connected. The authors prove that stereographic projection sends the boundary of this droplet to an ellipse in the plane. They also obtain an explicit mother body for the droplet by reducing the problem to an equilibrium problem with a weakly admissible field on the real line.

Core claim

When the two equal charges are sufficiently large the support of the equilibrium measure, called the droplet, is simply connected on the sphere. Its boundary, when mapped to the plane by stereographic projection, is an ellipse. A mother body for the droplet is obtained from an equilibrium problem with a weakly admissible external field on the real line.

What carries the argument

The droplet (support of the equilibrium measure on the sphere with external field from two point charges), whose boundary projects under stereographic projection to an ellipse.

If this is right

Beyond the transition the droplet fills the sphere except for a single connected complement instead of two separate caps.
The boundary curve admits an explicit description as the preimage of an ellipse under stereographic projection.
A mother body measure for the droplet can be constructed from a one-dimensional equilibrium problem on the line.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The ellipse description may permit closed-form expressions for the potential or energy associated with this configuration.
The same projection technique could be tested on equilibrium problems with three or more charges.
Locating the precise charge value at which connectedness changes would mark the topological transition explicitly.

Load-bearing premise

The analysis assumes that beyond some unspecified charge value the droplet is already simply connected, without locating the transition threshold or independently verifying connectedness.

What would settle it

A numerical or explicit calculation for a concrete charge value showing that the stereographic image of the boundary is not an ellipse or that the droplet still has holes.

Figures

Figures reproduced from arXiv: 1907.04801 by Arno B. J. Kuijlaars, Juan G. Criado del Rey.

read the original abstract

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by two equal point charges. The support of the equilibrium measure is known as the droplet. Brauchart et al. showed that the complement of the droplet consists of two spherical caps when the charges are small. When the charges are bigger the droplet becomes simply connected and we prove that the boundary of the droplet is mapped by stereographic projection to an ellipse in the plane. Moreover, we compute a mother body for the droplet that we derive from an equilibrium problem with a weakly admissible external field on the real line.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that for large equal charges the droplet boundary projects to an ellipse and gives an explicit mother body from a real-line problem, but assumes the simply-connected regime without locating the transition.

read the letter

The new piece is the ellipse description of the boundary under stereographic projection once the droplet turns simply connected, plus the mother-body construction obtained by reducing to a weakly admissible equilibrium problem on the line. This complements the two-cap result of Brauchart et al. for small charges by handling the opposite regime with a concrete geometric object and an explicit auxiliary measure. The reduction step looks useful because it turns a spherical problem into a one-dimensional one that can be solved more directly. The abstract presents both the ellipse claim and the mother body as theorems, so the work supplies at least the statements and the overall strategy. The main limitation is that the transition to the simply-connected case is stated without an explicit charge threshold or an independent check that connectedness holds above some value and that no extra singularities appear exactly at the change-over. The abstract simply says “when the charges are bigger” and moves to the ellipse result, so anyone wanting to apply the shape description still needs to know where the regime starts. The citation pattern is standard for this corner of potential theory and does not rely on circular arguments. This is a narrow but concrete advance inside logarithmic potential theory on the sphere. Readers who already work on equilibrium measures with point charges or on mother-body constructions will find the explicit ellipse and the line reduction worth seeing. It is solid enough to send to a referee who knows the Brauchart et al. paper and the relevant potential-theory tools; the proofs can be checked for gaps in the reduction and the connectedness step can be clarified if needed.

Referee Report

2 major / 2 minor

Summary. The paper examines the equilibrium measure (droplet) on the unit sphere in the presence of an external field from two equal point charges. It recalls the result of Brauchart et al. that the complement consists of two spherical caps for small charges, then asserts that for larger charges the droplet becomes simply connected; in that regime it proves that stereographic projection maps the droplet boundary to an ellipse in the plane and constructs an associated mother body by reduction to a one-dimensional equilibrium problem with a weakly admissible external field on the real line.

Significance. If the ellipse characterization and mother-body construction hold, the work supplies an explicit geometric description of the droplet in the simply-connected regime and a concrete link between the spherical problem and a classical one-dimensional mother-body problem. This would extend the small-charge results and provide a rare case of an explicitly solvable two-charge equilibrium problem on the sphere.

major comments (2)

[Introduction / §3] Introduction and §3 (setup of the simply-connected regime): the central ellipse theorem and mother-body derivation are stated only under the assumption that the droplet is simply connected for charges larger than some unspecified threshold. No derivation of the critical charge value is given, nor is an independent argument supplied showing that the support remains connected (and free of additional singularities) above that threshold. Because the ellipse claim is conditional on this regime, the gap is load-bearing for the main result.
[§4] §4 (ellipse proof): the reduction via stereographic projection to an ellipse appears to rely on the simply-connected assumption to eliminate the two-cap complement; without an a-priori verification that connectedness holds exactly when the external-field strength exceeds the (unspecified) transition point, it is unclear whether the ellipse description remains valid at the boundary of the regime or if additional topological changes intervene.

minor comments (2)

[Throughout] Notation for the external field strength (denoted variously as Q or t in different sections) should be unified and the dependence on the charge magnitude made explicit in all statements of the main theorems.
[§5] The mother-body construction in §5 invokes a weakly admissible external field on the line; a brief comparison with the classical admissible case (e.g., reference to the relevant theorem in the cited literature) would clarify why weak admissibility suffices here.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for the careful reading and the identification of load-bearing assumptions in the manuscript. We respond to the major comments below.

read point-by-point responses

Referee: [Introduction / §3] Introduction and §3 (setup of the simply-connected regime): the central ellipse theorem and mother-body derivation are stated only under the assumption that the droplet is simply connected for charges larger than some unspecified threshold. No derivation of the critical charge value is given, nor is an independent argument supplied showing that the support remains connected (and free of additional singularities) above that threshold. Because the ellipse claim is conditional on this regime, the gap is load-bearing for the main result.

Authors: The manuscript states the ellipse and mother-body results under the assumption that the droplet is simply connected for sufficiently large charges, motivated by the two-cap configuration established by Brauchart et al. for small charges. No derivation of the transition threshold or independent connectedness proof is supplied, as the paper's focus is the explicit characterization once the regime is entered. We will revise the introduction and §3 to state the conditional nature of the theorems more explicitly and to note that locating the transition remains outside the scope of the present work. revision: partial
Referee: [§4] §4 (ellipse proof): the reduction via stereographic projection to an ellipse appears to rely on the simply-connected assumption to eliminate the two-cap complement; without an a-priori verification that connectedness holds exactly when the external-field strength exceeds the (unspecified) transition point, it is unclear whether the ellipse description remains valid at the boundary of the regime or if additional topological changes intervene.

Authors: The argument in §4 uses the simply-connected hypothesis to guarantee that the complement is a single connected set, so that the projected boundary is a single closed curve shown to be an ellipse. The result is not asserted at the precise transition value, where the topology changes. We will add a clarifying remark in §4 specifying that the ellipse description applies for charges strictly larger than the (unspecified) transition point at which simple connectedness holds. revision: partial

standing simulated objections not resolved

Derivation of the critical charge value at which the droplet transitions from two caps to simply connected.
Independent argument establishing that the support remains connected and free of additional singularities for all charges above the transition threshold.

Circularity Check

0 steps flagged

No circularity; derivation uses external citation and standard machinery under stated regime assumption

full rationale

The paper cites Brauchart et al. (external) for the small-charge two-cap case, then states that for larger charges the droplet is simply connected and proves the stereographic ellipse boundary plus mother-body construction via potential-theoretic arguments. No step reduces a claimed prediction or theorem to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The regime assumption is explicit but does not create a definitional loop or force the ellipse result by construction; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on the standard definition of logarithmic equilibrium measure on the sphere and on the small-charge droplet description from prior literature; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

standard math Existence and uniqueness of the equilibrium measure minimizing the energy functional with external field on the sphere
Invoked implicitly when defining the droplet as the support of the equilibrium measure.
domain assumption For small charges the complement of the droplet consists of two spherical caps (Brauchart et al.)
Used as the baseline from which the large-charge regime is distinguished.

pith-pipeline@v0.9.0 · 5628 in / 1254 out tokens · 23434 ms · 2026-05-24T23:14:23.957556+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

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F. Balogh, M. Bertola, S.-Y. Lee, and K. D. T-R McLaughlin, Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane , Comm. Pure Appl. Math. 68 (2015), no. 1, 112–172

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J. S. Brauchart, P. D. Dragnev, and E. B. Saﬀ, Riesz external ﬁeld problems on the hypersphere and optimal point separation, Potential Analysis 41 (2014), no. 3, 647–678

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J. S. Brauchart, P. D. Dragnev, E. B. Saﬀ, and R. S. Womersley,Logarithmic and Riesz equilibrium for multiple sources on the sphere: the exceptional case , Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan (J. Dick, F. Kuo, and H. Wo´ zniakowski, eds.), Springer, Cham, 2018, pp. 179–203

work page 2018
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V. S. Buyarov and E. A. Rakhmanov, Families of equilibrium measures in an external ﬁeld on the real axis , Sb. Math. 190 (1999), no. 5-6, 791–802, translation from Mat. Sb. 190 (1999), no. 6, 11–22

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Chavel, Eigenvalues in Riemannian Geometry , Pure and Applied Mathematics, vol

I. Chavel, Eigenvalues in Riemannian Geometry , Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984

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Deift, T

P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external ﬁeld , J. Approx. Theory 95 (1998), no. 3, 388–475

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P. D. Dragnev, On the separation of logarithmic points on the sphere , Approxima- tion Theory X: Abstract and Classical Analysis (C. K. Chui, L. L. Schumaker, and J. Stoeckler, eds.), Vanderbilt University Press, Nashville, TN, 2002, pp. 137–144

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P. D. Dragnev and E. B. Saﬀ,Riesz spherical potentials with external ﬁelds and minimal energy points separation, Potential Anal. 26 (2007), no. 2, 139–162

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O. Frostman, Potentiel d’´ equilibre et capacit´ e des ensembles. Avec quelques applications a la th´ eorie des fonctions, Thesis, Lund, Imprimerie H˚ akan Ohlsson, 1935. 25

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A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of ratio- nal approximation of analytic functions , Math. USSR. Sb. 62 (1989), no. 2, 305–348, translation from Mat. Sb. 134(176) (1987), No.3, 306–352

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Gustafsson and J

B. Gustafsson and J. Roos, Partial balayage on Riemannian manifolds , J. Math. Pures Appl. (9) 118 (2018), 82–127

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A. B. J. Kuijlaars and G. L. F. Silva, S-curves in polynomial external ﬁelds , J. Approx. Theory 191 (2015), 1–37

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Mart´ ınez-Finkelshtein, R

A. Mart´ ınez-Finkelshtein, R. Orive, and E. A. Rakhmanov, Phase transitions and equilibrium measures in random matrix models, Comm. Math. Phys. 333 (2015), no. 3, 1109–1173

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Mart´ ınez-Finkelshtein and E

A. Mart´ ınez-Finkelshtein and E. A. Rakhmanov,Critical measures, quadratic diﬀeren- tials, and weak limits of zeros of Stieltjes polynomials , Comm. Math. Phys. 302 (2011), no. 1, 53–111

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, Do orthogonal polynomials dream of symmetric curves? , Found. Comput. Math. 16 (2016), no. 6, 1697–1736

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Orive, J

R. Orive, J. F. S´ anchez Lara, and F. Wielonsky,Equilibrium problems in weakly admis- sible external ﬁelds created by point charges , J. Approx. Theory 244 (2019), 71–100

work page 2019
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E. B. Saﬀ and V. Totik, Logarithmic Potentials with External Fields , Grundlehren der mathematischen Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997

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Simeonov, A weighted energy problem for a class of admissible weights , Houston J

P. Simeonov, A weighted energy problem for a class of admissible weights , Houston J. Math. 31 (2005), no. 4, 1245–1260. 26

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[1] [1]

Balogh, M

F. Balogh, M. Bertola, S.-Y. Lee, and K. D. T-R McLaughlin, Strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane , Comm. Pure Appl. Math. 68 (2015), no. 1, 112–172

work page 2015

[2] [2]

Beltr´ an,Harmonic properties of the logarithmic potential and the computability of elliptic Fekete points , Constr

C. Beltr´ an,Harmonic properties of the logarithmic potential and the computability of elliptic Fekete points , Constr. Approx. 37 (2013), no. 1, 135–165

work page 2013

[3] [3]

J. S. Brauchart, P. D. Dragnev, and E. B. Saﬀ, Riesz external ﬁeld problems on the hypersphere and optimal point separation, Potential Analysis 41 (2014), no. 3, 647–678

work page 2014

[4] [4]

J. S. Brauchart, P. D. Dragnev, E. B. Saﬀ, and R. S. Womersley,Logarithmic and Riesz equilibrium for multiple sources on the sphere: the exceptional case , Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan (J. Dick, F. Kuo, and H. Wo´ zniakowski, eds.), Springer, Cham, 2018, pp. 179–203

work page 2018

[5] [5]

V. S. Buyarov and E. A. Rakhmanov, Families of equilibrium measures in an external ﬁeld on the real axis , Sb. Math. 190 (1999), no. 5-6, 791–802, translation from Mat. Sb. 190 (1999), no. 6, 11–22

work page 1999

[6] [6]

Chavel, Eigenvalues in Riemannian Geometry , Pure and Applied Mathematics, vol

I. Chavel, Eigenvalues in Riemannian Geometry , Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984

work page 1984

[7] [7]

J. G. Criado del Rey, On the separation distance of minimal Green energy points on compact Riemannian manifolds, arXiv:1901.00779

work page internal anchor Pith review Pith/arXiv arXiv 1901

[8] [8]

Deift, T

P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external ﬁeld , J. Approx. Theory 95 (1998), no. 3, 388–475

work page 1998

[9] [9]

P. D. Dragnev, On the separation of logarithmic points on the sphere , Approxima- tion Theory X: Abstract and Classical Analysis (C. K. Chui, L. L. Schumaker, and J. Stoeckler, eds.), Vanderbilt University Press, Nashville, TN, 2002, pp. 137–144

work page 2002

[10] [10]

P. D. Dragnev and E. B. Saﬀ,Riesz spherical potentials with external ﬁelds and minimal energy points separation, Potential Anal. 26 (2007), no. 2, 139–162

work page 2007

[11] [11]

Frostman, Potentiel d’´ equilibre et capacit´ e des ensembles

O. Frostman, Potentiel d’´ equilibre et capacit´ e des ensembles. Avec quelques applications a la th´ eorie des fonctions, Thesis, Lund, Imprimerie H˚ akan Ohlsson, 1935. 25

work page 1935

[12] [12]

A. A. Gonchar and E. A. Rakhmanov, Equilibrium distributions and degree of ratio- nal approximation of analytic functions , Math. USSR. Sb. 62 (1989), no. 2, 305–348, translation from Mat. Sb. 134(176) (1987), No.3, 306–352

work page 1989

[13] [13]

Gustafsson and J

B. Gustafsson and J. Roos, Partial balayage on Riemannian manifolds , J. Math. Pures Appl. (9) 118 (2018), 82–127

work page 2018

[14] [14]

Gustafsson, R

B. Gustafsson, R. Teodorescu, and A. Vasil’ev, Classical and Stochastic Laplacian Growth, Advances in Mathematical Fluid Mechanics, Birkh¨ auser/Springer, Cham, 2014

work page 2014

[15] [15]

Hardy and A

A. Hardy and A. B. J. Kuijlaars, Weakly admissible vector equilibrium problems , J. Approx. Theory 164 (2012), no. 6, 854–868

work page 2012

[16] [16]

A. B. J. Kuijlaars and G. L. F. Silva, S-curves in polynomial external ﬁelds , J. Approx. Theory 191 (2015), 1–37

work page 2015

[17] [17]

Mart´ ınez-Finkelshtein, R

A. Mart´ ınez-Finkelshtein, R. Orive, and E. A. Rakhmanov, Phase transitions and equilibrium measures in random matrix models, Comm. Math. Phys. 333 (2015), no. 3, 1109–1173

work page 2015

[18] [18]

Mart´ ınez-Finkelshtein and E

A. Mart´ ınez-Finkelshtein and E. A. Rakhmanov,Critical measures, quadratic diﬀeren- tials, and weak limits of zeros of Stieltjes polynomials , Comm. Math. Phys. 302 (2011), no. 1, 53–111

work page 2011

[19] [19]

, Do orthogonal polynomials dream of symmetric curves? , Found. Comput. Math. 16 (2016), no. 6, 1697–1736

work page 2016

[20] [20]

Orive, J

R. Orive, J. F. S´ anchez Lara, and F. Wielonsky,Equilibrium problems in weakly admis- sible external ﬁelds created by point charges , J. Approx. Theory 244 (2019), 71–100

work page 2019

[21] [21]

E. B. Saﬀ and V. Totik, Logarithmic Potentials with External Fields , Grundlehren der mathematischen Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997

work page 1997

[22] [22]

Simeonov, A weighted energy problem for a class of admissible weights , Houston J

P. Simeonov, A weighted energy problem for a class of admissible weights , Houston J. Math. 31 (2005), no. 4, 1245–1260. 26

work page 2005