A lower Bound for the Area of Plateau Foams

J. M. Sotoca; S. Markvorsen; V. Gimeno

arxiv: 1906.08537 · v2 · pith:DQSRB5OUnew · submitted 2019-06-20 · 🧮 math.DG · physics.app-ph

A lower Bound for the Area of Plateau Foams

V. Gimeno , S. Markvorsen , J. M. Sotoca This is my paper

Pith reviewed 2026-05-25 19:29 UTC · model grok-4.3

classification 🧮 math.DG physics.app-ph

keywords Plateau foamssurface area lower boundsdivergence theoremKelvin foamminimal periodic foamsPlateau singularitiesfoam geometry

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The pith

A modified divergence theorem for foams with Plateau singularities provides lower bounds on their surface areas.

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

The paper develops a lower bound for the surface area of a spherically bounded piece of foam. It applies a new version of the divergence theorem that is adapted to the geometry of foams and accounts for the singularities at Plateau borders. This produces explicit estimates for the fundamental cell of a Kelvin foam, for a cost function, and for pressure differences in minimal periodic foams. An algorithm is also given that accepts a set of isolated points and returns the best lower bound for any foam having those points as vertices.

Core claim

By means of a divergence theorem modified to handle the classical Plateau singularities, the surface area of a foam piece inside a sphere is bounded from below in terms of volume and boundary contributions, and this bound is applied to obtain concrete numerical estimates for the area of the fundamental cell in a Kelvin foam together with related quantities for periodic minimal foams.

What carries the argument

The adapted divergence theorem for foams, which integrates vector fields while respecting the triple junctions and edge singularities prescribed by Plateau's laws.

If this is right

Lower bounds for the area of the fundamental cell of a Kelvin foam.
Lower bounds for the cost function associated to foams.
Lower bounds for pressure differences in minimal periodic foams.
An algorithm that computes the best lower bound from any prescribed finite set of vertex points.

Where Pith is reading between the lines

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

The bound could be used to test whether a given foam configuration is close to area-minimizing.
The same technique might supply area estimates for foams whose topology differs from the Kelvin cell.
Numerical implementation of the algorithm could generate candidate minimal foams by optimizing vertex positions.
The method might extend to foams whose films carry additional curvature constraints beyond the Plateau laws.

Load-bearing premise

The modified divergence theorem correctly accounts for the geometry at the classical Plateau singularities where three films meet.

What would settle it

Calculate the actual surface area of a known Kelvin foam cell inside a sphere and check whether the value lies above the lower bound produced by the theorem for the same spherical domain.

Figures

Figures reproduced from arXiv: 1906.08537 by J. M. Sotoca, S. Markvorsen, V. Gimeno.

**Figure 1.** Figure 1: Cutting off spherical scoops – extrinsic foam discs – from affine foams around an edge point (top row) and a vertex point (bottom row), respectively [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Piece of the Kelvin foam and an (enlarged) extrinsic disc of the Kelvin foam centered at a vertex point. at every vertex of the given foam in order to achieve an effective lower bound for the total area of a foam. We will assume throughout that the length of the mean curvature vector of the faces of the foam F is bounded by a constant h as follows: (1) |H(x)| ≤ h for all x ∈ F . Our main objective is to de… view at source ↗

**Figure 3.** Figure 3: In this case, Theorem 1.1 says that for any point [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 3.** Figure 3: From left to right: cutting off extrinsic foam discs from a catenoid, a cylinder and a sphere 2. Outline of the Paper In section 3 we shall introduce all required propositions and preliminaries, including the adapted version of the divergence theorem, needed to prove the main theorem of the paper in section 4. The last part of the paper, section 5, is devoted to show some applications of the main theorem.… view at source ↗

**Figure 4.** Figure 4: Double bubble foam. Hence we obtain Rmax ≥ 1 h Finally by using the main theorem, A(F) ≥ A(Dmax) ≥ A(D1/h) ≥ θ(o) e 2 π h 2 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Domain DΩ(X) from example 5.1. has no critical points in DΩ(X)∩ B∞ rmax \ {~0}, the maximum of the definition of the extrinsic vertex area is attained in the boundary eva(X, Ω, h) = max ~r∈∂(DΩ(X)∩B∞rmax ) (X N i=1 πe−2hri θvr 2 i ) . and hence, evA(X, Ω, h) = max Z⊂X [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: (a) We show the solution that maximizes eva for the case of eight vertices inside the sphere domain ∂Ω. (b) We show the distribution of spheres that maximizes evA Algorithm 1 evA calculation 1: Input: 2: X: set of N-isolated points into Ω ⊂ R 3 with boundary ∂Ω. 3: Output: 4: evA = 0. 5: loop: 6: Estimate the distance matrix DΩ(X) in X with boundary ∂Ω. 7: Build the set of inequalities M.r ≤ b (see 68). 8:… view at source ↗

read the original abstract

Real foams can be viewed as a geometrically well-organized dispersion of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at the Plateau borders according to the celebrated Plateau's laws. In this paper we estimate from below the surface area of a spherically bounded piece of a foam. Our main tool is a new version of the divergence theorem which is adapted to the specific geometry of a foam with special attention to its classical Plateau singularities. As a benchmark application of our results we obtain lower bounds for the fundamental cell of a Kelvin foam, lower bounds for the so-called cost function, and for the difference of the pressures appearing in minimal periodic foams. Moreover, we provide an algorithm whose input is a set of isolated points in space and whose output is the best lower bound estimate for the area of a foam that contains the given set as its vertex set.

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 adapted divergence theorem for Plateau foams is the core new item, but its validity at triple lines and quadruple points determines whether the lower bounds hold.

read the letter

The paper introduces a version of the divergence theorem adjusted for foam surfaces that meet along curves at 120 degrees and at quadruple points, then uses it to produce explicit lower bounds on the area of a spherically bounded foam piece. They also give an algorithm that takes a finite set of points as vertices and returns a lower bound on the area of any foam containing those vertices. As a test case they compute numbers for the Kelvin cell and for pressure differences in periodic foams. That combination of a modified integral identity plus a concrete computational procedure is what is actually new here relative to the cited literature on minimal foams. The algorithm in particular is a clear, implementable output that could be checked independently. The applications to the cost function and pressure gaps are straightforward once the theorem is granted. The central uncertainty is exactly the one flagged in the stress test: whether the new divergence identity correctly captures the contributions from the singular 1-skeleton or whether extra Hausdorff-measure terms or jump conditions are missing. The abstract states the theorem but does not reproduce the derivation or the verification at the singularities, so the numerical bounds for the Kelvin cell rest on an unexamined step. If that step is handled correctly the bounds are useful modest results; if not, the estimates are unsupported. The work is aimed at geometric analysts and materials modelers who need area lower bounds or quick vertex-based estimates rather than full minimal-surface solutions. It is coherent on its own terms and engages the standard Plateau laws, so it is worth sending to referees even though the theorem proof will need close checking.

Referee Report

1 major / 2 minor

Summary. The paper claims to derive lower bounds on the surface area of a spherically bounded piece of a Plateau foam by means of a new divergence theorem adapted to the foam's singular geometry (including triple lines at 120° and quadruple points). Benchmark applications include explicit lower bounds for the fundamental cell of a Kelvin foam, the cost function, pressure differences in minimal periodic foams, and an algorithm that takes isolated points as input and outputs a lower bound on the area of any foam having those points as vertices.

Significance. If the adapted divergence theorem is valid, the work supplies concrete, computable lower bounds that can be used to test conjectures on minimal foams and to bound geometric quantities in periodic minimal surfaces. The algorithmic component offers a practical tool for vertex-constrained area estimation.

major comments (1)

[statement and proof of the adapted divergence theorem (abstract and introduction)] The central lower-bound claims rest on the new divergence theorem adapted to foams. The manuscript must supply a complete, self-contained proof (with explicit handling of the 1-skeleton) that the integral identity holds when the vector field is integrated across surfaces meeting along curves where three sheets meet at 120° and at points where four edges meet; any omitted Hausdorff-measure term on the singular set or incorrect jump condition would invalidate the subsequent Kelvin-cell and cost-function estimates.

minor comments (2)

Notation for the foam's singular strata (triple lines, quadruple points) should be introduced once and used consistently; currently the same symbols appear to be reused for both the geometric sets and their measures.
The algorithm description would benefit from a short pseudocode block or explicit complexity statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and the positive evaluation of the significance of the work. The principal concern is the need for a complete, self-contained proof of the adapted divergence theorem. We address this point below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

Referee: [statement and proof of the adapted divergence theorem (abstract and introduction)] The central lower-bound claims rest on the new divergence theorem adapted to foams. The manuscript must supply a complete, self-contained proof (with explicit handling of the 1-skeleton) that the integral identity holds when the vector field is integrated across surfaces meeting along curves where three sheets meet at 120° and at points where four edges meet; any omitted Hausdorff-measure term on the singular set or incorrect jump condition would invalidate the subsequent Kelvin-cell and cost-function estimates.

Authors: We agree that a complete, self-contained proof of the adapted divergence theorem is required to rigorously support the lower-bound claims. In the revised manuscript we will insert a dedicated section containing the full proof. The argument will explicitly treat the 1-skeleton, derive the integral identity across surfaces meeting at 120° along triple lines, and handle the quadruple points, confirming the appropriate jump conditions and verifying that no additional Hausdorff-measure contributions arise on the singular set. This expansion will also be referenced in the abstract and introduction as needed. revision: yes

Circularity Check

0 steps flagged

No circularity; lower bounds derived from independently established divergence theorem

full rationale

The paper's central derivation begins with a stated new version of the divergence theorem adapted to Plateau singularities (abstract and introduction), which is presented as the main tool and is then applied to obtain area lower bounds for spherically bounded foams and the Kelvin cell. No equations or steps reduce the final bounds to fitted parameters, self-referential definitions, or load-bearing self-citations whose content is unverified within the paper. The algorithm for point-set inputs likewise produces estimates via the theorem without circular renaming or ansatz smuggling. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard divergence theorem in smooth domains plus an unstated assumption that the adaptation to Plateau singularities preserves the integral identity; no free parameters or new physical entities are introduced in the abstract.

axioms (1)

standard math Divergence theorem holds for smooth vector fields on domains with piecewise smooth boundary
Base case invoked before the foam-specific adaptation.

pith-pipeline@v0.9.0 · 5711 in / 1133 out tokens · 29329 ms · 2026-05-25T19:29:43.742643+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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J. Taylor, The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces, Annals of Mathematics, Second Series, Vol. 103, No. 3 (May, 1976), pp. 489-539. 1Departament de Matem`atiques-IMAC, Universitat Jaume I, Castell ´o, Spain., 2DTU Compute, Technical University of Denmark., 3Departamento de Lenguajes y Sistemas Inform´aticos-I...

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[1] [1]

W. K. Allard, On the First Variation of a Varifold , Annals of Mathematics, 95 (1972), 417– 491

work page 1972

[2] [2]

Aref and D

H. Aref and D. L. Vainchtein, The equation of state of a foam , Phys. Fluids 12 (2000), no. 1, 23–28

work page 2000

[3] [3]

M. A. Fortes, The surface energy of ﬁnite clusters of soap bubbles , Phys. Fluids 13 (2001), no. 12, 3542–3546

work page 2001

[4] [4]

M. A. Fortes, The surface energy of strained foam clusters , Philosophical magazine letters 83 (2003), no. 2, 143–148

work page 2003

[5] [5]

M. A. Fortes, F Morgan, and M F´ atima Vaz, Pressures inside bubbles in planar clusters , Philosophical magazine letters 87 (2007), no. 8, 561–565. A LOWER BOUND FOR THE AREA OF PLATEAU FOAMS 19

work page 2007

[6] [6]

Hutchings, F

M. Hutchings, F. Morgan, M. Ritor´ e, and A. Ros, Proof of the Double Bubble Conjecture , Annals of Mathematics, 155 (2002), 459–489

work page 2002

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Michael Kleder, CON2VERT - constraints to vertices , http://www.mathworks.com/matlabcentral/ﬁleexchange/7894-con2vert-constraints-to- vertices

work page

[8] [8]

Rob Kusner, The number of faces in a minimal foam , Proc. Roy. Soc. London Ser. A 439 (1992), no. 1907, 683–686

work page 1992

[9] [9]

Sullivan, Comparing the Weaire-Phelan equal-volume foam to Kelvin’s foam, Forma 11 (1996), no

Rob Kusner and John M. Sullivan, Comparing the Weaire-Phelan equal-volume foam to Kelvin’s foam, Forma 11 (1996), no. 3, 233–242

work page 1996

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Lee, Introduction to smooth manifolds, second ed., Graduate Texts in Mathematics, vol

John M. Lee, Introduction to smooth manifolds, second ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013

work page 2013

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Morgan, Geometric Measure Theory: A Beginner’s Guide

F. Morgan, Geometric Measure Theory: A Beginner’s Guide . Academic Press, third ed., Aug. 2000

work page 2000

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Robert Osserman, A survey of minimal surfaces , second ed., Dover Publications Inc., New York, 1986

work page 1986

[13] [13]

10, 48–57

Sydney Ross, Bubbles and foams – new general law , Industrial & Engineering Chemistry 61 (1969), no. 10, 48–57

work page 1969

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149, American Mathematical Society, Providence, RI, 1996, Translated from the 1992 Japanese original by the author

Takashi Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996, Translated from the 1992 Japanese original by the author

work page 1996

[15] [15]

Taylor, The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces, Annals of Mathematics, Second Series, Vol

J. Taylor, The Structure of Singularities in Soap-Bubble-Like and Soap-Film-Like Minimal Surfaces, Annals of Mathematics, Second Series, Vol. 103, No. 3 (May, 1976), pp. 489-539. 1Departament de Matem`atiques-IMAC, Universitat Jaume I, Castell ´o, Spain., 2DTU Compute, Technical University of Denmark., 3Departamento de Lenguajes y Sistemas Inform´aticos-I...

work page 1976