{"id":"1ff38f78-b74e-4cd2-98da-33c8a20bce4d","arxiv_id":"2505.11650","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For small angular momentum, there exists a unique rotation orbit of smooth rotating capillary drop solutions near the circle, built by a variational bifurcation argument from a multiple eigenvalue.","lead":"This paper writes the free-boundary equations of a two-dimensional capillary liquid drop in Craig-Sulem form on the circle and then on the flat torus, proves their Hamiltonian structure, and shows that rotating wave solutions bifurcate from the static circle even without symmetry assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1 is stated for ℓ*≥1, but the bifurcation argument requires ℓ*≥2; at ℓ*=1, ω*=0, the quadratic angular-momentum term vanishes and the transversality condition (4.57) fails, so the theorem as stated is false.","rationale":"The paper's main theorem is a local bifurcation result. Read in good faith, the Lyapunov-Schmidt construction is coherent for the intended case ℓ*≥2: the kernel is the two-dimensional VN, the linearized operator is invertible on the complement, and the reduced angular momentum is nondegenerate. I therefore do not doubt the strategy. The most load-bearing defect is in the statement: (4.24) permits ℓ*=1, and every nondegeneracy used to solve for ω(v) and reparametrize by a vanishes at ℓ*=1. This is not a stylistic issue: the asserted √a estimate fails in that case. The fix is minimal (add ℓ*≥2), which supports a conditional acceptance rather than rejection. The star-shaped graph ansatz is not a serious concern here, because the constructed solutions are small perturbations of the circle and hence stay in that class; the sign/factor inconsistencies in F0 and (3.24) are also real but appear to be transcription errors and do not affect the F=∇E system actually used in the proof. The reader's conditional verdict is therefore appropriate, but their weakest_assumption (overhangs) is not the load-bearing gap; the ℓ*≥2 condition is.","tokens_in":36102,"tokens_out":36742,"duration_ms":349457,"concrete_test":"Set ℓ*=1 in (4.24) and recompute (4.70): I0(v)=0 for every v∈VN. Then examine Lemma 4.16(ii): the fixed-point equation (4.79)-(4.80) with I_N((1+µ)Λv)=|v|² has no solution with bounded µ, since the leading term (2µ+µ²)|v|² would have to balance a cubic remainder. Independently, check that the branch predicted by the proof would satisfy I≈O(|v|³), contradicting (4.4). A direct way to settle the concern is to add the hypothesis ℓ*≥2 and verify that (4.57) and (4.70) then give strictly positive coefficients.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equation (4.24) allows ℓ*=1, which gives ω*=0. The proof of the orbit parametrized by a relies on two steps that both break down in this case. First, Lemma 4.14 solves F(ω,v)=0 for ω=ω(v) by the implicit function theorem using (4.54), ∂ωΦ(ω*,0,y) = -1/2|ΠZN J0∂ϑ y|². By (4.57), this quantity contains the factor 4ω*²ℓ*²(1+ω*²)^{-1}, so it vanishes identically for ℓ*=1. Second, Lemma 4.16 uses the leading quadratic term I0(v)=ω*ℓ*(1+ω*²)^{-1}(|v1|²+|v−1|²) from (4.70) to construct the diffeomorphism ψ with I_N(ψ(v))=|v|²; for ℓ*=1 this coefficient is zero, and the angular momentum of the reduced branch is at least cubic in v, incompatible with the bound |ωa−ω*|+...≤C√a in (4.4). Thus Theorem 4.1 as written is not valid for ℓ*=1. The defect is not in the strategy: for ℓ*≥2 the kernel is two-dimensional, ω*>0, and the same proof gives the claimed family. The statement should assume ℓ*≥2, and a similar condition is needed in Theorem 4.2(i) and in the ω* formula (4.98).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":36456,"tokens_out":16646,"duration_ms":146429,"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":[{"comment":"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.","section":"§4.3, Theorem 4.1 and (4.24)"},{"comment":"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.","section":"§4.1, Eq. (4.11)"},{"comment":"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).","section":"§3.2, Eq. (3.24)"},{"comment":"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.","section":"§4.4, Lemma 4.19"}],"minor_comments":[{"comment":"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.","section":"§4.3, Theorem 4.1"},{"comment":"The proof refers to 'Sections 3.2 and 3.3', but the relevant material is in Sections 4.2 and 4.3.","section":"§4.3, Proof of Theorem 4.1"},{"comment":"The proof cites 'Lemma 4.35', but no such lemma exists; the intended reference is presumably Lemma 4.12.","section":"§4.3, Lemma 4.17(iii)"},{"comment":"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.","section":"§4.4, Proof of Theorem 4.2(iii)"}],"recommendation":"major_revision","confidential_remarks":"The ℓ*=1 gap is not a cosmetic issue: it invalidates Theorem 4.1 and Theorem 4.2(i) exactly as stated. However, the intended argument is sound for ℓ*≥2, and the sign/display errors in (4.11) and (3.24) are localized and correctable. If the authors restrict the main theorems to ℓ*≥2 and fix the displayed identities, the paper is likely acceptable. I would ask the authors to also double-check the transversality statement in Lemma 4.19, since the current proof verifies a different condition."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nI agree with the reader's conditional verdict, but I want to sharpen it: the flaw is not just that the proof is sketchy—it is that Theorem 4.1 as stated includes ℓ*=1, where the argument provably breaks down. At ℓ*=1, ω*=0, so the transversality condition in Lemma 4.14 (equation (4.57)) is identically zero, and the implicit function theorem that selects ω(v) cannot be applied. The same degeneracy kills the quadratic leading term I0(v) in Lemma 4.16, so the angular momentum cannot be reparametrized to give the claimed √a bound. The proof works for ℓ*≥2, where ω*>0 and the factor is nonzero. This is a real bug, but a localized one: adding ℓ*≥2 to Theorem 4.1, Theorem 4.2(i), and to the ω* formula in (4.98) fixes it.\n\nThe paper's actual contributions are solid: the torus version of the Craig-Sulem equations with the extra curvature terms, the Hamiltonian structure, and the variational bifurcation from the 3-dimensional kernel using torus equivariance. The kernel computation is careful and the use of the Tα-action is clean. The author is honest that Theorem 4.2(iii) recovers the Moon–Wu result only locally.\n\nThere are also internal inconsistencies that the authors should fix: the abstract equations (1.14) have a sign in the capillary term that does not match F0 in (4.11), and the gradient formula (3.24) is missing the e^ξ factors that appear in (4.14). These are fixable but they make the paper hard to check. The proof of Theorem 4.1 is just a two-line delegation to earlier lemmas; after adding the ℓ*≥2 hypothesis, that proof needs to be written out carefully.\n\nThis paper is for researchers in free-boundary problems and nonlinear bifurcation theory. The strategy is sound for the corrected statement, and the formulation work is worth having. I would send it to a serious referee, with the expectation of a revision that fixes the ℓ*=1 gap and the notational discrepancies.","headline":"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.","tokens_in":36964,"tokens_out":5978,"would_cite":false,"duration_ms":53056,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","35B32","35C07","76B45","35B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["capillary liquid drop","free boundary problem","Craig-Sulem formulation","Dirichlet-Neumann operator","rotating waves","bifurcation from multiple eigenvalues","angular momentum","Hamiltonian structure"],"falsifier":"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 $|v|^2=a$.","tokens_in":35912,"feed_emoji":"💧","tokens_out":9120,"duration_ms":90330,"temperature":0.7,"pith_summary":"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.","feed_headline":"One rotating-wave orbit per angular momentum for a 2D drop","feed_subtitle":"A Hamiltonian boundary formulation plus rotation symmetry turns a kernel of dimension three into one analytic family of rigidly rotating…","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the boundary normal-derivative operator estimates, symmetry properties, shape derivative, and analyticity used throughout the functional setup.","marker":"[38]"},{"why":"Provides the finite-dimensional reduction for Hamiltonian systems near an equilibrium that the author adapts to the torus action.","marker":"[44]"},{"why":"Gives the normal-modes result behind parametrizing solutions by conserved angular momentum.","marker":"[56]"},{"why":"Supplies the variational bifurcation-from-multiple-eigenvalues strategy for traveling capillary-gravity waves.","marker":"[22]"},{"why":"Establishes the prior existence of two-dimensional rotating waves under reversibility and c-fold symmetry; the local reversible/c-fold result here recovers and extends it.","marker":"[43]"},{"why":"Gives the three-dimensional capillary-drop formulation and Hamiltonian structure whose two-dimensional analogue is derived here.","marker":"[11]"},{"why":"Proves bifurcation from multiple eigenvalues for rotating traveling waves on the three-dimensional drop; the present proof follows its reduction scheme.","marker":"[12]"},{"why":"Introduced the formally rotating nearly circular drop and motivates the rotating-wave ansatz.","marker":"[47]"}],"fun_headline_variants":["Symmetry turns triple eigenvalue into unique rotating-wave orbit","2D capillary drop: rotation symmetry yields unique wave orbits","For every angular momentum, a single rotating wave orbit","Multiple eigenvalue bifurcation: one orbit per angular momentum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry turns triple eigenvalue into unique rotating-wave orbit","2D capillary drop: rotation symmetry yields unique wave orbits","For every angular momentum, a single rotating wave orbit","Multiple eigenvalue bifurcation: one orbit per angular momentum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000261,"raw_usage":{"total_tokens":1676,"prompt_tokens":1111,"completion_tokens":565,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":727,"completion_tokens_details":{"reasoning_tokens":501}},"tokens_in":727,"tokens_out":565,"duration_ms":5483,"temperature":1.0,"reasoning_tokens":501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:51:42.226740+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $|v|^2=a$.","supporting_citations":[{"cited_title":"Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics , American Mathematical Soc., (2013)","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary normal-derivative operator estimates, symmetry properties, shape derivative, and analyticity used throughout the functional setup."},{"cited_title":"Moser , Periodic orbits near an equilibrium and a theorem by Alan Weinstein , Comm","cited_arxiv_id":null,"evidence_quote":"Provides the finite-dimensional reduction for Hamiltonian systems near an equilibrium that the author adapts to the torus action."},{"cited_title":"Weinstein, Normal modes for nonlinear hamiltonian systems , Invent","cited_arxiv_id":null,"evidence_quote":"Gives the normal-modes result behind parametrizing solutions by conserved angular momentum."},{"cited_title":"Craig, D","cited_arxiv_id":null,"evidence_quote":"Supplies the variational bifurcation-from-multiple-eigenvalues strategy for traveling capillary-gravity waves."},{"cited_title":"Baldi, D.A","cited_arxiv_id":null,"evidence_quote":"Proves bifurcation from multiple eigenvalues for rotating traveling waves on the three-dimensional drop; the present proof follows its reduction scheme."},{"cited_title":"Rayleigh, On the capillary phenomenon of jets , Proc","cited_arxiv_id":null,"evidence_quote":"Introduced the formally rotating nearly circular drop and motivates the rotating-wave ansatz."}],"review_version":1}