Rigidity for capillary liquid drops of nearly circular section with constant vorticity

Domenico Angelo La Manna; Giuseppe La Scala; Pietro Baldi

REVIEW 2 major objections 5 minor 26 references

Rigidity for capillary liquid drops of nearly circular section with constant vorticity

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that any steady rotating capillary drop whose equatorial section is C²-close to a disc must be an oblate spheroid rotating as a solid body, provided the vorticity–capillarity–size parameter stays below 64/3.

desk verdict Rigidity threshold extension for near-circular constant-vorticity drops is real, but Theorem 1.1 overclaims δ-uniformity; add an η-gap and it's solid. read the letter →

arxiv 2607.17844 v1 pith:35HKUA3I submitted 2026-07-20 math.AP

Pietro Baldi , Domenico Angelo La Manna , Giuseppe La Scala This is my paper · ORCID

classification math.AP MSC 35R3576B4535B0635N25

keywords capillaryliquiddropconstantvorticityfreeboundaryproblemrigidityshapefunctionalsDirichlet–NeumannoperatornearlycirculardomainTaylorexpansion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a rigidity theorem for steady three-dimensional capillary liquid drops with constant vorticity. If the drop's equatorial section is C²-close to a disc and the parameter α0²/σ0 (|D|/π)^{3/2} is below 64/3, then the drop must be an oblate spheroid and the flow must be uniform rotation about the vertical axis. This greatly widens the range where rigidity was known: an earlier variational proof only covered the threshold √2, and the new perturbation argument reaches a value about fifteen times larger. The proof expands a shape functional that vanishes on every solution, shows its quadratic part is positive on all nonconstant Fourier modes precisely in this parameter range, and controls the cubic remainder by regularity bootstrapping. If the theorem holds, it rules out all non-axisymmetric steady drops in a C²-neighborhood of the disc below this threshold.

What carries the argument

Two ingredients carry the proof. First, an identity from the companion paper [4], σ0A(D)+(α0²/2)B(D)=0, where A and B are boundary integrals of the torsion function; after rescaling D to area π this becomes C=A+λ0B=0 with λ0=α0²/(2σ0)(|D|/π)^{3/2}. Second, the Dirichlet–Neumann operator of the nearly circular domain gives an analytic expansion of the boundary gradient of the torsion function; its second-order Taylor term in spherical-harmonic coefficients is, up to a constant, Σ_{ℓ≥2}ℓ(ℓ−1)(ℓ+2−3λ0/8)h²_{ℓ,m}. These coefficients are all positive exactly when λ0<32/3, and a bootstrap regularity lemma (also from [4]) bounds the cubic remainder by a small multiple of the quadratic term, forcing

What would settle it

Compute the functional C=A+λ0B for a family of area-π domains with small elliptic perturbation, e.g., h=ε cos(2θ) with ε≪1. If C vanishes for some ε≠0 and λ0<32/3, the theorem is false; a numerical evaluation of the four boundary integrals defining A and B for ellipses of various eccentricities would settle this.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for any α0,σ0>0 there exists δ>0 such that every C² steady solution of the free-boundary Euler equations with constant vorticity, whose equatorial section D has barycenter at the origin and whose boundary is the graph of h over the unit circle with ∥h∥_{C²}<δ, is rigid whenever α0²/σ0 (|D|/π)^{3/2}<64/3. Rigid here means D is a disc, the drop is the oblate spheroid with profile (1.4),(1.11), flattened at the poles and bulging at the equator, and the velocity is u=½α0(−x2,x1,0). The previous rigidity threshold from variational methods was α0²/σ0 (|D|/π)^{3/2}<√2; the new result extends rigidity up to 64/3, a roughly fifteenfold increase, for nearly circular s

Load-bearing premise

The proof depends on two results imported from the authors' companion paper [4] — the overdetermined identity (1.15) and the bootstrap estimate ∥h∥_{C⁶}≤C∥h∥_{C²} (Lemma 2.1) — and if either fails for C² solutions in the enlarged parameter range, the rigidity conclusion has no foundation.

Editorial extensions

If this is right

If Theorem 1.1 is correct, no steady non-axisymmetric rotating drop with a nearly circular equatorial section exists for α0²/σ0 (|D|/π)^{3/2}<64/3; the only possible shape is the oblate spheroid.
The previous variational threshold √2 is not optimal; the new perturbation threshold 64/3 is a genuine improvement for C²-nearly-circular sections.
At the boundary value 64/3, the quadratic form degenerates on the ℓ=2 Fourier mode, so the method identifies the natural place where non-circular solutions might first appear.
The proof gives an explicit positive lower bound on the quadratic form in H^{3/2}, yielding a quantitative estimate of how far a nearly circular solution must be from the disc if it is not a solution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The theorem's δ is claimed to depend only on α0 and σ0, but the proof requires δ to be comparable to the coercivity constant µ0(λ0), which shrinks to zero as λ0→32/3; therefore the uniformity of δ over all domains satisfying (1.14) is not established by the argument.
At λ0=32/3 the quadratic term vanishes at mode ℓ=2, so one expects a bifurcation of ellipsoidal drops; computing the fourth-order term of C near the disc would test whether such solutions exist and whether the threshold is sharp.
The Taylor-expansion-plus-coercivity scheme generalizes: any overdetermined problem that can be written as a functional identity with a positive quadratic part on high modes and a controlled remainder admits the same rigidity argument, so the method may apply to other free-boundary symmetry problems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a legitimate perturbation result that pushes the rigidity threshold for nearly circular capillary drops from λ0 < √2/2 to λ0 < 32/3, and the mechanism is transparent. The one real problem is the uniformity claim for δ: as written, Theorem 1.1 says δ depends only on α0 and σ0, but the proof forces δ ≤ c(32/3 − λ0), and λ0 can approach 32/3 for fixed α0, σ0 by varying |D|. So the statement overclaims. The fix is to put in an explicit gap λ0 ≤ 32/3 − η and let δ depend on η.

What's new: the second-order expansion of the functional C = A + λ0 B around the unit disc, the explicit diagonalization giving positive coefficients for all modes ℓ ≥ 2 when λ0 < 32/3, and the handling of the ℓ=0,1 modes through area and barycenter constraints. The Taylor lemmas are detailed and the logic from expansion to coercivity to rigidity is sound. The authors are also honest that the starting identity (1.15), the regularity bootstrap, and analyticity come from their companion preprints [2] and [4]. That is a self-citation burden but not a circularity: the cited results do the heavy lifting, and this paper's contribution is the expansion and the positivity argument.

Soft spots: the δ-uniformity issue above is the only serious one I see. It doesn't sink the method but it does mean Theorem 1.1 as stated is not fully supported. A second, minor point: the proof leans on [4] for smoothness of the boundary, stated as Lemma 2.1 with a brief proof sketch. A referee will want to check that the bootstrap actually gives the C6 control with a constant independent of the particular solution in the δ-neighborhood. Probably fine, but it deserves a look. The lengthy algebraic expansions in Lemmas 2.5–2.8 are not machine-checked; I didn't spot an error, but a referee should verify a few of those integrations.

Who this is for: people working on free-boundary Euler equations with vorticity, symmetry/rigidity problems, or shape perturbation for overdetermined problems. It deserves a serious referee, and the main result is likely correct after the uniformity repair.

Referee Report

2 major / 5 minor

Summary. The paper studies stationary solutions of the free-boundary Euler equations for a three-dimensional capillary liquid drop with constant vorticity. The main result (Theorem 1.1) states that if the equatorial section D of a C^2 solution has its barycenter at the origin, its boundary is a C^2-small graph over the unit circle, and the dimensionless parameter λ0 = (α0^2/(2σ0))(|D|/π)^{3/2} is below 32/3, then D must be exactly a disc, the drop is an oblate spheroid with profile (1.4),(1.11), and the velocity is the rigid rotation (α0/2)(-x2,x1,0). The proof normalizes the area, expands the shape functional C=A+λ0B around the unit disc using the Dirichlet-Neumann operator, and shows that for λ0<32/3 the quadratic part is coercive on the nonconstant modes. The paper includes detailed Taylor expansions and a coercivity argument.

Significance. If valid, the result extends the known rigidity threshold from the variational bound (1.10), which corresponds to λ0<√2/2, to λ0<32/3 under the additional near-circularity assumption. This is a substantial quantitative improvement and the conclusion is falsifiable: it predicts the nonexistence of non-axisymmetric nearly circular steady drops in that parameter range. The paper contains a new expression for the boundary gradient of the torsion function in terms of the Dirichlet-Neumann operator (Lemma 2.2), and the Taylor expansions are carried out with care. No fitted parameters are used. However, the main theorem overclaims the uniformity of δ, and the proof relies on unpublished preprints for essential ingredients.

major comments (2)

[Theorem 1.1 and eq. (2.39)] The proof does not establish the stated dependence of δ on α0,σ0 only. After normalizing |D'|=π, the coercivity constant μ0 defined below (2.37) is actually μ0 = (π/16)(32−3λ0) for λ0<32/3 (the minimum is attained at ℓ=2). The argument requires ∥h∥_{C^2}<δ with Cδ ≤ μ0/2, so δ must be ≤ C^{-1}μ0 = O(32/3−λ0). Since λ0 depends on |D| and condition (1.14) only imposes λ0<32/3, a drop with area arbitrarily close to the threshold forces μ0→0 and hence δ→0. Thus no single δ depending only on α0,σ0 can cover all admissible areas. The theorem should be amended, e.g., by requiring λ0 ≤ 32/3 − η and taking δ=δ(α0,σ0,η), or by letting δ depend on λ0.
[Identity (1.15) and Lemma 2.1] The proof is not self-contained at load-bearing points. Identity (1.15) is the starting point of the Taylor analysis, and Lemma 2.1 supplies the high-order regularity used in the interpolation step (2.41). Both are imported from the companion preprint [4], and Lemma 2.3 uses the analyticity result of preprint [2]. Since these are cited as preprints, the present paper does not establish the foundation of Theorem 1.1 on its own. The authors should either include the necessary statements/proofs or clarify that the result is conditional on [2] and [4].

minor comments (5)

[Abstract] The phrase 'we show that a rigidity result holds also above the threshold' should be qualified: the result requires near-circularity and a barycenter normalization; without this, the abstract could suggest a fully global result.
[Eq. (2.10)] The expression g'(0) = ∇_{S^1}h/2 + (Gh)−h/2 x is ambiguous; parentheses would help, e.g., g'(0) = (1/2)∇_{S^1}h + ((Gh)−h)/2 x.
[Reference [20]] 'hil. Mag.' should read 'Phil. Mag.'.
[Lemma 2.2] 'We denote G(h)ψ the Dirichlet-Neumann operator' should read 'We denote by G(h)ψ the Dirichlet-Neumann operator'.
[Section 2, after (2.37)] The term 'spherical harmonics' for eigenfunctions on S^1 is unconventional; 'trigonometric polynomials' or 'Fourier basis' would be clearer.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the conclusion is not inserted into the hypotheses. The main caveat is a self-citation burden — the key identity (1.15), the regularity bootstrap, and the final determination of the spheroid are imported from the authors' own companion preprints [4] and [2] — but this is provenance dependence, not a circular reduction.

full rationale

Theorem 1.1 is not circular in the decisive sense: the paper proves C(D') = 0 from the algebraic balance identity (1.15), then shows by an independent Taylor expansion and a coercivity estimate that C(D') ≥ (μ0/2)||a||^2_{H^{3/2}} for small h, with equality only when all nonzero Fourier modes of h vanish. The coercive term (2.37) is computed from Lemmas 2.4–2.8 using the Dirichlet–Neumann formula (Lemma 2.2); it is not fitted and does not assume the conclusion. The area normalization and barycenter hypothesis are used honestly to eliminate the ℓ=0 and ℓ=1 modes, and the cubic remainder is controlled by Lemma 2.1; none of these steps defines the target shape into the assumptions. The cited identity (1.15) is derived in [4] from the solution structure, not from disc rigidity, so it is independent support even though [4] is a companion preprint by the same authors. Likewise, the regularity bootstrap and the analyticity of the Dirichlet–Neumann operator are parameter-free regularity tools whose stated assumptions do not include the rigidity conclusion. There is no fitted parameter renamed as a prediction and no uniqueness theorem invoked to force the answer. The real weaknesses are correctness/self-containedness concerns rather than circularity: the proof leans on [4] and [2] for load-bearing steps, and the uniformity of δ in Theorem 1.1 appears to require δ to shrink as λ0 approaches 32/3, since μ0 = O(32/3 − λ0). This is a gap in the stated uniformity claim, not a circular step, and it is repairable by adding a positive margin λ0 ≤ 32/3 − η. Overall, no circular reduction is exhibited; the score reflects the moderate self-citation burden, not a derivational circle.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters and no invented entities. The proof assumes the solution-dependent identity (1.15) and a bootstrap lemma from the authors' own [4], plus analyticity from the authors' own [2]; near-circularity provides convexity and star-shapedness. These are domain assumptions from companion work, not ad hoc to this paper.

assumptions (4)

domain assumption For every steady solution, the normalized identity C(D)=A(D)+λ0B(D)=0 holds with λ0=(α0²/2σ0)(|D|/π)^{3/2}.
Entered at (1.15)–(1.18), derived in [4] from the free-boundary Euler equations; the central input of the paper.
domain assumption If D is C²-close to a disc, then D is convex and its boundary admits the graph representation γ(x)=x(1+h(x)).
Used in (1.13) and in Lemma 2.1; true for δ small, but convexity is not explicitly restated in Theorem 1.1.
domain assumption Bootstrap regularity estimate ∥h∥_{C6(S¹)} ≤ C∥h∥_{C2(S¹)} for solutions (Lemma 2.1).
Cited to [4]; essential for the Sobolev interpolation step in (2.41).
domain assumption Analyticity of the Dirichlet–Neumann map G(h) and of φ(h)=|(∇v)∘γ|, with cubic Taylor remainder (Lemma 2.3 and (2.3)–(2.4)).
Appeals to the authors' preprint [2, Thm 1.1]; supplies the Taylor expansion and remainder estimates for the shape functionals.

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Pith. "Pith review of Rigidity for capillary liquid drops of nearly circular section with constant vorticity." pith.science (2026). https://pith.science/paper/35HKUA3I

@misc{pith2026260717844,
  author       = {Pith},
  title        = {Pith review of: Rigidity for capillary liquid drops of nearly circular section with constant vorticity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35HKUA3I}},
  note         = {Machine review of arXiv:2607.17844}
}

read the original abstract

We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity. A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [2]

Baldi, V

P. Baldi, V. Julin, D.A. La Manna , On the Dirichlet-Neumann operator for nearly spherical domains, preprint, arXiv:2603.27865

arXiv

[2] [4]

Baldi, D.A

P. Baldi, D.A. La Manna, G. La Scala , A rigidity result for the 3D capillary liquid drop with constant vorticity, preprint, arXiv:2607.00450

arXiv

[3] [1]

Baldi, V

P. Baldi, V. Julin, D.A. La Manna , Liquid drop with capillarity and rotating traveling waves, Arch. Rational Mech. Anal. 250 (2026), article no.\,4

2026

[4] [3]

P. Baldi. D.A. La Manna, G. La Scala , Bifurcation from multiple eigenvalues of rotating traveling waves on a capillary liquid drop, preprint, arXiv:2504.01555

arXiv

[5] [5]

Barbieri, M

T. Barbieri, M. Berti, A. Maspero, M. Mazzucchelli , Bifurcation of gravity-capillary Stokes waves with constant vorticity, J.\ Differential Equations 451 (2026)

2026

[6] [6]

Berti, L

M. Berti, L. Franzoi, A. Maspero , Traveling quasi-periodic water waves with constant vorticity, Arch.\ Ration.\ Mech.\ Anal.\ 240 (2021), 99-202

2021

[7] [7]

Chandrasekhar , The Stability of a Rotating Liquid Drop, Proceedings of the Royal Society of London

S. Chandrasekhar , The Stability of a Rotating Liquid Drop, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, May 25, 1965, Vol. 286, No. 1404 (May 25, 1965), pp. 1-26

1965

[8] [8]

Europhys

A.\ Constantin, E.\ Kartashova , Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. Europhys. Lett. 86 (2009), 29001

2009