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Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves

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Pith's one-line read A nearly circular two-dimensional capillary drop admits, for each sufficiently small angular momentum, exactly one orbit of rigidly rotating wave solutions, analytic in the momentum.

desk verdict The main theorem is stated too broadly: at ℓ*=1 the transversality and the quadratic angular-momentum term vanish, so Theorem 4.1 is false as written, but the intended result is sound for ℓ*≥2 and the paper's formulation work is valuable. read the letter →

arxiv 2505.11650 v1 pith:AYBDEDUF submitted 2025-05-16 math.AP

Giuseppe La Scala This is my paper · ORCID
classification math.AP MSC 35R3535B3235C0776B4535B38
keywords capillaryliquiddropfreeboundaryproblemCraig-SulemformulationDirichlet-NeumannoperatorrotatingwavesbifurcationfrommultipleeigenvaluesangularmomentumHamiltonianstructure
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 establishes that a two-dimensional, incompressible, irrotational liquid drop held together by surface tension and shaped nearly as a circle can rotate rigidly in a one-parameter family of solutions: for each sufficiently small value $a$ of the angular momentum there is exactly one orbit of rotating-wave profiles, and the whole orbit varies analytically with $a$. To reach that conclusion, the free-boundary Euler problem is first rewritten as an equivalent system on the flat one-dimensional torus, with a boundary normal-derivative operator modified by conformal and curvature factors. The system is shown to be Hamiltonian, and the conserved angular momentum arising from translation invariance is used to parametrize solutions. The linearization at the static circle has a three-dimensional kernel at resonant frequencies, so the classical simple-eigenvalue bifurcation theorem does not apply; the paper instead reduces the bifurcation equation by the torus action and obtains a unique orbit of rotating waves for each small momentum. If correct, this gives a rigorous existence and uniqueness statement for rotating drops without imposing reversibility or discrete symmetry in advance.

What carries the argument

The load-bearing object is the conformal parametrization of the drop boundary: the map $(\rho,\vartheta)\mapsto e^\rho(\cos\vartheta,\sin\vartheta)$ sends the infinite periodic strip to the punctured plane, and the drop surface is written as $\rho=\xi(\vartheta)$ with $\xi=\log(1+h)$. Pulling back the free-boundary problem gives a capillary water-wave-type system on the one-dimensional torus whose principal operator is the torus Dirichlet-Neumann operator $G(\xi)$ (the map sending boundary potential to the normal derivative of its harmonic extension), plus explicit curvature terms. The variational machinery is the functional $E=H-\omega(I-a)$: its critical points on the level set $I=a$ are exactly rotating waves, the torus action $T_\alpha$ acts by translation of the angle and preserves $E$, and the conserved angular momentum $I(\xi,\chi)=-\tfrac12\int e^{2\xi}\chi'\,d\vartheta$ (equivalently $\int e^{2\xi}\xi'\chi\,d\vartheta$) provides the parameter $a$. A finite-dimensional reduction with the torus action collapses the three-dimensional kernel to the one-dimensional quotient, and a reparametrization of the momentum level sets turns the constraint $I=a$ into a circle $|v|^2=a$ on which the reduced critical-point problem has a single orbit.

What would settle it

Fix $\sigma_0=1$, take the resonant mode $\ell_*=2$ (so $\omega_*=\sqrt{3/2}$), and solve the torus equations numerically with initial data a small multiple of the two oscillatory kernel modes and angular momentum $I=a$. The theorem predicts that all sufficiently small time-periodic solutions with $I=a$ lie on one analytic orbit up to rotation; finding two distinct, non-rotationally related profiles with the same $a$ and the same $\omega$ would refute the uniqueness claim. Analytically, one could also compute the next coefficient in the expansion $\omega(a)=\omega_*+c\sqrt a+\cdots$; the theorem's bound $|\omega_a-\omega_*|\le C\sqrt a$ requires consistency, while a second branch would appear as a second zero of the reduced equation on the circle $|v|^2=a$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the symmetry of the problem is strong enough to replace the missing transversality of a multiple eigenvalue. Setting $\xi=\log(1+h)$ and $\chi$ as the boundary potential pulled back to the flat torus, the drop equations take the form $\partial_t\xi=e^{-2\xi}G(\xi)\chi$ together with a companion equation for $\partial_t\chi$, with Hamiltonian $H$ and conserved angular momentum $I$. At angular velocities $\omega_*=\sqrt{\sigma_0}\sqrt{\ell_*^2-1}/\ell_*$ chosen so that one Fourier mode $\ell_*$ is resonant, the linearized operator has kernel spanned by three vectors, one constant and two oscillatory modes of mode $\ell_*$. Instead of relying on a simple eigenvalue, the author uses the invariance of the functional $H-\omega(I-a)$ under the rotation group $T_\alpha$ to reduce the bifurcation equation to a finite-dimensional problem on the two-dimensional oscillatory kernel; solving that reduced equation produces, for every small $a$, a single $T_\alpha$-orbit of rotating waves with $I=a$, analytic in $a$ and with $|\omega_a-\omega_*|+\|\eta_a\|+\|\beta_a\|\le C\sqrt a$. Theorem 4.2 then shows that within the same orbit one may choose a symmetric representative (even $\eta$, odd $\beta$), and that imposing $c$-fold symmetry yields the same uniqueness statement; thus every orbit is generated by a symmetric rotating wave.

Load-bearing premise

The entire reduction assumes the drop boundary is a star-shaped graph of the form $\partial\Omega_t=\{(1+h(t,x))x: x\in S^1\}$ with $1+h>0$; if the free boundary develops an overhang or self-intersection, the elevation-and-boundary-potential formulation no longer describes the system, so the rotating-wave branches are only established inside this star-shaped class.

Editorial extensions

If this is right

  • For every small enough $a$, the rotating-wave family is unique up to rigid rotation, so observing a second non-translated rotating drop profile with the same angular momentum would contradict the theorem.
  • No discrete symmetry needs to be assumed a priori: reversibility and $c$-fold symmetry emerge as properties of a representative of the unique orbit, not as restrictions imposed to make existence possible.
  • The torus formulation brings the two-dimensional capillary drop into the same form as capillary water-wave equations, so the existing regularity and perturbation results for that form become applicable to the drop problem.
  • The analytic dependence on $a$ gives a rigorous small-amplitude expansion $\omega(a)=\omega_*+O(\sqrt a)$, $\eta,\beta=O(\sqrt a)$, which can be compared with numerical continuation from each linear mode $\ell_*$.
  • The same orbit uniqueness is available with $c$-fold symmetry imposed, and the symmetric representative of each orbit is the one generated by an even-velocity-potential, odd-potential profile.

Reading between the lines

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

  • The quotient-by-torus-action reduction is likely to work in other settings where the linearized kernel has dimension larger than one: as long as the angular momentum is nondegenerate and the kernel splits into a single rotation pair, one expects one orbit per momentum rather than isolated waves.
  • An analogous statement may hold for the three-dimensional capillary drop, with the rotation group acting on spherical harmonics; the precise resonant frequencies would involve spherical harmonic degrees instead of Fourier modes $\ell_*$.
  • A testable prediction is the leading asymptotic law $\omega(a)-\omega_*\propto\sqrt a$ with a coefficient computable from the cubic terms of the reduced functional; Fourier-space simulations of the torus equations could check whether all time-periodic solutions at fixed small $I=a$ collapse to a single orbit.
  • Because the formulation is Hamiltonian with a conserved momentum, the rotating waves are natural candidates for persistence under small time-periodic or non-autonomous perturbations, though the paper does not discuss such persistence.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the free boundary problem for a two-dimensional, pure capillary, irrotational incompressible drop whose boundary is a star-shaped graph over the unit circle. It derives a Craig–Sulem formulation on S^1 and then on the flat torus T^1 via the conformal parametrization of the exterior of the origin, proves the Hamiltonian structure of the torus equations, and identifies conserved quantities from translation and shift symmetries, in particular the angular momentum. The main result, Theorem 4.1, claims that for every small positive value a of the angular momentum there is a unique T^1-orbit of rotating-wave solutions bifurcating from the static circle, with the wave speed and Sobolev norms controlled by C√a, and that the orbit depends analytically on a. Theorem 4.2 gives analogous statements for solutions with reversibility symmetry, c-fold symmetry, and both. The proof uses a Lyapunov–Schmidt reduction with a two-dimensional kernel, an implicit-function solution of the range equation, a choice of ω = ω(v) from a quadratic bifurcation equation, and a reparametrization of the angular-momentum constraint by |v|², following the Moser–Weinstein/Craig–Nicholls approach.

Significance. If the corrections below are made, the paper would be a valuable contribution to the rigorous theory of capillary drops: it gives the first Craig–Sulem formulation for the 2D drop on the torus, a clean Hamiltonian framework, and a self-contained bifurcation argument that parametrizes rotating waves by angular momentum rather than by a fitted parameter. The explicit computation of the linearized operator, the detailed Lyapunov–Schmidt reduction, and the use of the torus equivariance to obtain uniqueness of the orbit are genuine strengths. The geometric transformations in Section 2 are carefully derived, and the analyticity and tame estimates for the Dirichlet–Neumann operator are invoked from the water-wave literature in a reasonable way. The main claims are, however, stated too broadly: the case ℓ*=1 is included in the theorems but breaks the bifurcation argument, and several displayed equations that are load-bearing for the formulation are mutually inconsistent. These issues are localized and appear repairable by restricting to ℓ*≥2 and correcting the displayed formulas, so the underlying strategy is defensible.

major comments (4)
  1. [§4.3, Theorem 4.1 and (4.24)] The statement allows ℓ*=1, for which ω*=0, and the proof cannot work in that case. Lemma 4.14 solves the bifurcation equation by the implicit function theorem using the transversality quantity ∂ωΦ(ω*,0,y0) = -1/2|ΠZN J0∂ϑ y0|². By (4.57) this quantity contains the factor 4ω*²ℓ*²(1+ω*²)^{-1}, so it vanishes identically when ℓ*=1 and the implicit function theorem is unavailable. In the same way, the leading quadratic term I0(v) in (4.70) is proportional to ω*ℓ*(1+ω*²)^{-1}, so for ℓ*=1 the angular momentum of the reduced branch is at least cubic in v; Lemma 4.16 cannot construct the diffeomorphism ψ with IN(ψ(v))=|v|², and the bound |ωa−ω*|+‖ηa‖+‖βa‖≤C√a in (4.4) cannot hold. Thus Theorem 4.1 is false as stated for ℓ*=1. The theorem and its proof should assume ℓ*≥2, and Theorem 4.2(i), which uses the same ω*, needs the same restriction.
  2. [§4.1, Eq. (4.11)] The zero-set of the operator F0 defined in (4.10)-(4.11) does not agree with the rotating-wave equation obtained from the Craig–Sulem system. From (1.10), with the ansatz ξ(t,ϑ)=η(ϑ+ωt), χ(t,ϑ)=β(ϑ+ωt), the second component should be F2,0 = ωβ′ − e^{−2η}½((G(η)β+η′β′)/√(1+η′²))² + e^{−2η}½β′² − σ0(e^{−η}[η′/√(1+η′²)]′ − e^{−η}/√(1+η′²)+1). Equation (4.11) has a plus sign before the σ0 bracket, which is the opposite sign. Consequently Lemma 4.4's assertion that F0(ω;η,β)=0 characterizes solutions of the ansatz is not correct as written. The later reduction uses F=∇E, which is consistent with (1.10), so the error is localized, but the displayed equations in the introduction and in Section 4.1 must be reconciled.
  3. [§3.2, Eq. (3.24)] The displayed formula for the L²-gradient ∂ξH is missing the factors e^ξ (and e^{2ξ}) in the curvature term. The computation in the proof of Lemma 3.4(ii) and the Hamiltonian (1.11) give −σ0(e^ξ[ξ′/√(1+ξ′²)]′ − e^ξ/√(1+ξ′²)+e^{2ξ}), which is the expression later used in (4.14), not the printed −σ0([ξ′/√(1+ξ′²)]′ − 1/√(1+ξ′²)+e^{2ξ}). As printed, (3.24) does not yield the Hamiltonian equations (3.22) claimed in Lemma 3.4(ii).
  4. [§4.4, Lemma 4.19] The transversality check in part (i) is misdirected. The Crandall–Rabinowitz condition required in part (ii) is ∂²_{ωu}F(ω*,0)[v1] ∉ R, where for the F used in this section ∂²_{ωu}F(ω*,0)[v1] = −J0∂ϑv1. Part (i) only proves ∂ϑv1 ∈ V = Z, which is a different statement. For ℓ*=1, ω*=0, v1=(cosϑ,0), and −J0∂ϑv1=(0,sinϑ) lies in the range R, so the transversality condition actually fails. For ℓ*≥2 the condition can be verified directly, but the proof as written does not establish it, and the argument for Theorem 4.2(i) must be rewritten accordingly.
minor comments (4)
  1. [§4.3, Theorem 4.1] The statement repeats the regularity assumption: it first says 'Let s≥0, s0>0' and then 'Let s≥s0>1'; this should be cleaned up.
  2. [§4.3, Proof of Theorem 4.1] The proof refers to 'Sections 3.2 and 3.3', but the relevant material is in Sections 4.2 and 4.3.
  3. [§4.3, Lemma 4.17(iii)] The proof cites 'Lemma 4.35', but no such lemma exists; the intended reference is presumably Lemma 4.12.
  4. [§4.4, Proof of Theorem 4.2(iii)] The phrase 'we fix any ℓ*∈N' is slightly confusing because the c-fold analysis then uses frequencies cℓ*; this is fine mathematically but should be phrased more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rotating-wave bifurcation proof is self-contained; self-citations are contextual and non-load-bearing.

full rationale

The derivation chain is self-contained. The Craig-Sulem equations (1.7)-(1.8) are derived in Theorem 3.1 from the Euler equations (1.1)-(1.4) and the explicit star-shaped ansatz (1.5); the torus formulation (1.9)-(1.10) follows in Theorem 3.2 from the conformal map (2.24) and the identities in Lemma 2.2. The Hamiltonian (1.11) and the angular momentum (1.12) are computed from first principles in Lemmas 3.4-3.5, not posited as outputs to be matched. In Section 4 the rotating-wave equation is reduced via the explicit kernel/range decomposition (4.26)-(4.33); the range equation is solved by the Implicit Function Theorem in Lemma 4.11, the wave speed omega(v) is selected by solving the transversality/bifurcation equation (4.59) using (4.57), and Lemma 4.16 uses the reparametrization psi only as a change of variables on V_N, with I_N(psi(v)) = |v|^2, rather than as a fit of omega to target data. Lemmas 4.17-4.18 solve the reduced bifurcation equation and identify the T_alpha-orbit using the independent index theorem of Mawhin-Willem [41]. Citations to the author's own preprints [11,12] are contextual ('2D version ... treated in [11,12]') or stylistic ('proofs very similar to [12]'); the proofs are written out in the paper, and the substantive technical inputs (Dirichlet-Neumann estimates from Lannes and Berti-Maspero-Ventura, Crandall-Rabinowitz, Moser-Weinstein) are external. No equation is shown to equal its own input by construction, no fitted parameter is renamed as a prediction, and no load-bearing conclusion delegates to a self-citation. The l*=1 degeneracy noted in review is a possible correctness gap in the statement of Theorem 4.1, not a circularity.

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

The proof borrows standard results about the Dirichlet-Neumann operator (Lemma 2.4) and general bifurcation theorems; these are independent prior results, not invented here. The only modeling input beyond the Euler equations is the star-shaped graph representation and the analytic regularity framework. No constants are fitted to data, and no new physical entities are introduced.

assumptions (4)
  • domain assumption Dirichlet-Neumann analyticity, tame estimates, and shape derivative formulas (Lemma 2.4) are taken from [38, 39, 18].
    The implicit function theorem steps (Lemmas 4.9, 4.11, 4.13) rest on the analyticity and estimates for G(ξ); if these fail near zero, the Lyapunov-Schmidt reduction breaks.
  • domain assumption The free boundary is a star-shaped graph, ∂Ωt = {(1+h(t,x))x : x∈S1} with 1+h>0, and the conformal map ϕ in (2.24) is used to pass to the torus.
    The torus formulation and the Dirichlet-Neumann representation require the boundary to be a graph over S1; overhangs and self-intersections are excluded.
  • domain assumption The Bernoulli constant is chosen as c(t)=σ0, fixing the gauge of the velocity potential.
    The Hamiltonian (1.11) and the equation (1.10) depend on this choice; the paper notes it is a gauge freedom, but the chosen form is an assumption on the potential.
  • domain assumption The solution is sought in analytic spaces H^{s,s} with a smallness condition ||η||_{H^{s,s0+1}} < δ0 (Lemma 2.4, points (vii) and (ix)).
    Analytic regularity is required for the implicit function theorem and for the analytic dependence on the angular momentum; this restricts the size of the bifurcating waves.

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Pith. "Pith review of Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves." pith.science (2026). https://pith.science/paper/AYBDEDUF

@misc{pith2026250511650,
  author       = {Pith},
  title        = {Pith review of: Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbbT^1$ and bifurcations from multiple eigenvalues of rotating waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYBDEDUF}},
  note         = {Machine review of arXiv:2505.11650}
}
read the original abstract

We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in Baldi-Julin-La Manna [11] and Baldi-La Manna-La Scala [12]. In particular, we derive its Craig-Sulem formulation firstly over the circle, then over the one-dimensional flat torus; the arising equations are similar to the pure capillary Water Waves for the ocean problem, apart from conformal factors and additional terms due to curvature terms. Then, we show its Hamiltonian structure and we derive constants of motions from symmetries, one of which is the invariance by the torus action. Thanks to this invariance, we show the existence of orbits of rotating wave solutions (which are the analogous of travelling waves of the ocean problem) by bifurcation from multiple eigenvalues in the spirit of Moser-Weinstein [44, 56] and Craig-Nicholls [22] variational approaches; in particular, we can parametrize such orbits by the angular momentum, and for each value of it they are unique. This will imply that each orbit is generated by symmetric rotating waves.

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Forward citations

Cited by 2 Pith papers

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    For nearly circular capillary drops with constant vorticity, rigidity to the oblate-spheroid solution is shown for a new parameter range up to 64/3, well above the earlier variational threshold.

  2. Exponential Vorticity Hessian Growth in Capillary Liquid Drop in Two Dimensions

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    For the 2D free-boundary Euler equations with surface tension on a droplet, there exist arbitrarily small initial velocities such that the vorticity Hessian grows at least exponentially in time.

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