{"id":"34195017-5300-49a5-836e-cad437f51179","arxiv_id":"2504.14392","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A strictly convex capillary hypersurface with prescribed k-th Weingarten curvature exists whenever the prescribed data is symmetric under horizontal reflection.","lead":"This paper proves existence of convex bubble-like hypersurfaces in the upper half space whose kth curvature matches any given symmetric data. It is the capillary analogue of a classical theorem for closed convex surfaces, and it also gives a counterexample showing a natural balance condition is not necessary.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's C^0 estimate jumps from bounded capillary radii to a bound on ||h||_{C0}; the missing gauge lemma for capillary even solutions is the most load-bearing gap. The §5 expansion error is real but peripheral.","rationale":"The reader's weakest_assumption identifies capillary evenness as load-bearing because it fixes horizontal translation invariance in the support function equation. My review agrees and sharpens this into a concrete gap in the proof of Theorem 3.1: the displayed chain in the proof of Theorem 3.1 passes from ρ+(Σhat,θ)≤C to ||h||_{C^2}≤C without establishing the gauge-dependent C^0 bound. Since Eq. (1.3) is invariant under h↦h+Σ a_α⟨ξ,E_α⟩, no C^0 estimate can hold without a gauge condition, and capillary evenness is the only such condition available. The paper states in the introduction that evenness serves exactly this purpose, but the proof of Theorem 3.1 does not contain the required lemma. This is a fixable but central gap. The §5 expansion error is genuine: a direct expansion gives a_1=kH_1, not (n−k)H_1, so the counterexample proof of Theorem 1.3 is invalid as written; however Theorem 1.1 does not depend on Section 5. The θ=π/2 endpoint is also not covered by Theorem 3.1, though the introduction's reflection argument may handle it if made explicit. These issues justify keeping the reader's CONDITIONAL verdict and MODERATE confidence; the central existence theorem is plausible and the identified gaps are locally repairable, not demonstrably fatal.","tokens_in":18410,"tokens_out":26141,"duration_ms":243387,"concrete_test":"Add and prove the missing gauge lemma: for a capillary even admissible support function h, use h(ξhat)=h(ξ) and the reconstructing formula X(ξ)=∇_{S^n}h(ξ)+h(ξ)(ξ−cosθ e) to show X(ξhat)=R X(ξ), where R(x',x_{n+1})=(−x',x_{n+1}). Conclude the convex body is invariant under x'↦−x', so the horizontal center of any enclosing capillary ball is 0. Then verify that Σhat⊂C_{ρ+,θ}(0) implies |h(ξ)|≤Cρ+ for all ξ, which closes the inequality chain in the proof of Theorem 3.1. If this computation produces any residual horizontal drift, the C^0 estimate fails and Theorem 1.1 is unproved; if it succeeds, the concern is resolved. A useful cross-check is to apply the same argument to h_a(ξ)=ℓ(ξ)+a·ξ', which solves (1.3) with the same f but is not capillary even, and confirm that ρ+ stays bounded while ||h_a||_{C0}→∞.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence proof hinges on the a priori estimate Theorem 3.1. In its proof, after Lemma 2.3, Lemma 3.2 and Theorem 3.6, the text derives ρ+(Σhat,θ)^2 ≤ Cρ−(Σhat,θ) max λ_n ≤ C(1+ρ+(Σhat,θ)), and hence ρ+≤C. The second inequality uses max λ_n ≤ C(1+||h||_{C0}) together with ρ−≤C, and therefore implicitly uses an estimate ||h||_{C0} ≤ C(1+ρ+). But for the support function equation (1.3), horizontal translations are exact symmetries: if h solves (1.3), so does h_a(ξ)=h(ξ)+Σ_α a_α⟨ξ,E_α⟩, because the added horizontal first harmonic satisfies the Robin condition ∇_μ h_a = cotθ h_a on ∂Cθ. Thus a bound on capillary radii alone cannot control ||h||_{C0} without fixing this translation gauge. The only gauge condition available is capillary evenness, h(ξ)=h(ξhat), yet no lemma in Section 3 states or proves that evenness forces the horizontal translation to vanish and hence gives ||h||_{C0}≤C from ρ+≤C. This is precisely the step the introduction identifies as the role of capillary evenness, but the proof of Theorem 3.1 does not supply the needed argument. If this step cannot be justified, Theorem 3.1 fails and the degree-theoretic proof of Theorem 1.1 has a gap at its core. Separately, the expansion (5.2) in Section 5 is algebraically incorrect: for A_t=I+tV, expanding H_n(A_t)/H_{n−k}(A_t) gives a_1=kH_1, not (n−k)H_1. That error undermines Theorem 1.3 as written, but it does not affect the central existence theorem. The endpoint θ=π/2 is also excluded from Theorem 3.1 although Theorem 1.1 includes it; if the intended reflection argument is relied upon, it is not invoked in the proof of Theorem 1.1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper treats the prescribed k-th Weingarten curvature problem for strictly convex capillary hypersurfaces in the upper half-space R^{n+1}_+. It reformulates the problem as the Hessian quotient equation with Robin boundary condition (1.3) for the capillary support function h on Cθ, proves a priori C^{4,γ} estimates for admissible capillary even solutions (Theorem 3.1), and uses degree theory to obtain existence (Theorem 1.1), uniqueness near the constant solution (Theorem 1.2), and a counterexample showing that the capillary Minkowski condition (1.2) is not sufficient for 1≤k≤n−1 (Theorem 1.3). The main technical novelty is a quantitative bound relating the largest principal radius to the C0 norm of h, combined with a capillary version of Chou–Wang's geometric lemma.","tokens_in":18812,"tokens_out":20135,"duration_ms":196990,"significance":"The result is a natural and valuable extension of Guan–Guan's theorem for closed hypersurfaces and of the authors' capillary Minkowski result. The PDE formulation is clean, the degree-theoretic strategy is appropriate, and the paper correctly identifies capillary evenness as the translation-gauge condition. If the missing gauge argument is supplied, Theorem 1.1 would be a substantial contribution. The uniqueness theorem and the counterexample also add value. I am not convinced at present that Theorem 3.1 is proved as written, because the C0 estimate contains a load-bearing gap; the remaining issues are local and fixable.","major_comments":[{"comment":"The conclusion ρ_+(Σhat,θ)≤C does not follow from the displayed chain. The text writes ρ_+(Σhat,θ)^2 ≤ Cρ_-(Σhat,θ) max λ_n ≤ C(1+ρ_+(Σhat,θ)), but Theorem 3.6 supplies max λ_n ≤ C(1+‖h‖_{C0}), not C(1+ρ_+). This matters because horizontal translations are exact symmetries of (1.3): h_a(ξ)=h(ξ)+Σ_α a_α⟨ξ,E_α⟩ solves the same equation and the same Robin condition, while the capillary radii ρ_±(Σhat,θ) are unchanged. Thus capillary radii alone cannot control ‖h‖_{C0} without fixing the translation gauge. Capillary evenness is exactly the assumption that should fix the gauge, but no lemma in Section 3 states or proves that evenness yields ‖h‖_{C0}≤C(1+ρ_+), for instance by showing the body is contained in a fixed ball centered on the vertical axis. Without such a lemma, the a priori estimate and hence the degree-theoretic proof of Theorem 1.1 are incomplete.","section":"§3, proof of Theorem 3.1"},{"comment":"The existence of a contact point of ∂E_hatb with Σhat that is not on the flat boundary is asserted rather than proved. The sentence \"otherwise all the touch points lie in ∂R^{n+1}, and it is impossible\" does not explain why a maximizing ellipsoid in the capillary setting cannot have all its contact points on the supporting hyperplane. This point is needed for the inequality λ_{x,Σ}≥λ_{x,∂E_hatb} in (2.3) and therefore for the key estimate (1.6). Please provide a complete argument.","section":"§2.2, proof of Lemma 2.3"},{"comment":"Theorem 3.1 and Lemmas 3.4–3.5 are stated only for θ∈(0,π/2), while Theorem 1.1 includes θ=π/2. The proof of Theorem 1.1 invokes Theorem 3.1 without explaining how the endpoint is handled. If the reflection reduction mentioned in the introduction is intended to cover θ=π/2, it should be stated explicitly in the proof of Theorem 1.1; otherwise a separate or limiting argument is needed.","section":"Theorem 1.1 vs. Theorem 3.1"}],"minor_comments":[{"comment":"The displayed expansion is algebraically incorrect. For A_t=I+tV, the first-order coefficient is kH_1, not (n−k)H_1, and the H_2 coefficient in a_2 should be (n−k)(2n−k−1)/2 instead of (n−k)(n+k−1)/2. The conclusion of Theorem 1.3 is unaffected because the erroneous terms multiply functions whose first moment against ⟨ξ,E_α⟩ vanishes by (5.1), but the formula should be corrected.","section":"§5, Eq. (5.2)"},{"comment":"The displayed lower bound c_0(1+‖h‖_{C0})^{n−1}σ appears to have the wrong sign in the exponent. The determinant lower bound σ_n(A)≥c together with the upper bound λ_i≤C(1+‖h‖_{C0}) gives λ_i≥c(1+‖h‖_{C0})^{-(n−1)}, so the exponent should be −(n−1). Please correct.","section":"§3, Theorem 3.6"},{"comment":"The set B in the degree argument does not explicitly impose h>0, although Theorem 3.1 is stated for positive h. Please either include positivity in the definition of B or add a remark explaining that positivity follows from admissibility and the Robin condition.","section":"§4, definition of B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript leans on several companion papers by the same authors ([39], [40], [41]) for foundational facts, and on [28] for uniqueness of the constant-curvature capillary hypersurface; referees should verify the availability and precise statements of those results. The main theorem's central a priori estimate has a fixable but genuine gap in the C0 bound, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a natural open case: existence of strictly convex capillary hypersurfaces in the half-space with prescribed k-th Weingarten curvature, for 1 ≤ k ≤ n−1 and capillary even data. If the main theorem holds, it extends Guan-Guan's closed case and the authors' own capillary Minkowski result. The proof adapts known machinery—Chou-Wang, Lieberman–Trudinger, degree theory—and includes a genuinely new quantitative estimate: max principal radius is controlled linearly by the C^0 norm of the support function. That estimate is a real improvement over their earlier Minkowski paper.\n\nThe central a priori estimate (Theorem 3.1) is where the paper is currently softest. The proof derives ρ+^2 ≤ C ρ− max λ_n and then writes ≤ C(1+ρ+), which implicitly requires ||h||_{C0} ≤ C(1+ρ+). But horizontal translations are exact symmetries of the equation and of the capillary radii, so that bound is false without fixing the translation gauge. Evenness of h is the intended gauge, but no lemma states or proves that evenness forces the optimal cap center to the origin, which is what would give the missing C^0 bound. This is a load-bearing gap in the written proof. It looks fixable with a short argument, but as it stands Theorem 3.1 doesn't close.\n\nThere are also smaller issues. Lemma 2.3 asserts without proof that the maximal John ellipsoid touches the hypersurface at an interior point. The θ = π/2 endpoint is excluded from Theorem 3.1 although Theorem 1.1 claims it. And in §5 the expansion (5.2) has wrong coefficients—a_1 should be kH_1, and the H_2 term in a_2 is off. The counterexample's conclusion survives because the wrong terms integrate to zero by (5.1); so this is a cosmetic blemish, not a fatal one. The introduction also misstates Theorem 1.3: the constructed hypersurfaces show condition (1.2) is not necessary for solvability, not not sufficient.\n\nOverall, the main theorem is plausible and important, the estimates are mostly detailed and correct, and none of the gaps look deep. But the C^0 gauge issue needs to be fixed before the proof is fully trustworthy. I'd send this to a serious referee, and I'd expect a revision that adds the missing gauge lemma and cleans up the endpoint and counterexample details.","headline":"A solid and significant extension of the Guan–Guan existence theory to capillary hypersurfaces, but the proof has a fixable C^0 gauge gap that needs to be addressed before the main theorem is fully trustworthy.","tokens_in":19403,"tokens_out":17751,"would_cite":true,"duration_ms":144057,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C21","35B65","35J60","53C42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every positive smooth capillary-even function on a spherical cap is realized as the k-th Weingarten curvature of a strictly convex capillary hypersurface in the upper half-space, for any 1 ≤ k ≤ n−1 and contact…","keywords":["capillary hypersurfaces","Weingarten curvature","Hessian quotient equation","Robin boundary condition","degree theory","prescribed curvature problem","spherical cap","support function"],"falsifier":"A direct numerical or analytical continuation from $f\\equiv1$ along a path of capillary even functions, monitoring the eigenvalues of $\\nabla^2 h + h\\sigma$; if a smooth positive even $f$ with bounded $C^3$ norm produced a support function leaving a fixed $C^{4,\\gamma}$ ball or losing convexity, the a priori estimate of Theorem 3.1 and the degree-theoretic existence proof would be false.","tokens_in":18193,"feed_emoji":"📐","tokens_out":18115,"duration_ms":147270,"temperature":0.7,"pith_summary":"This paper asks whether a positive function prescribed on a spherical cap can be realized as the $k$-th Weingarten curvature of a strictly convex hypersurface that meets the boundary plane of the upper half-space at a fixed contact angle. The authors prove that the answer is yes for every smooth positive 'capillary even' function — one symmetric under horizontal reflection — for any $1 \\le k \\le n-1$ and contact angle $\\theta \\in (0, \\pi/2]$. The proof rephrases the geometric problem as a Hessian quotient equation with a Robin boundary condition for the support function, then solves that equation by a priori estimates and degree theory. The result extends the capillary Minkowski problem ($k = n$) to all intermediate curvature functions, and also yields uniqueness near constant data and a counterexample showing that the classical volume-balance condition is not necessary for $k < n$.","feed_headline":"Every symmetric curvature datum on a spherical cap is realized","feed_subtitle":"New proof gives convex half-space hypersurfaces with prescribed k-th Weingarten curvature.","key_machinery":"The load-bearing structure is the Hessian quotient equation (1.3), rewritten from the geometric prescription: $$\\frac{\\sigma_n(\\$nabla^{2}$ h + h\\$\\sigma$)}{\\sigma_{n-k}(\\$nabla^{2}$ h + h\\$\\sigma$)} = $f^{{-1}}$ \\quad \\text{in } C_\\$\\theta$, \\qquad \\nabla_\\mu h = \\cot\\$\\theta$\\, h \\quad \\text{on } \\partial C_\\$\\theta$,$$ where $h$ is the capillary support function on the spherical cap $C_\\theta$ and $\\sigma$ is the round metric. The matrix $A = \\nabla^2 h + h\\sigma$ has eigenvalues equal to the principal radii of the hypersurface, so admissibility ($A>0$) is exactly strict convexity. The argument runs on three mechanisms: a maximum-principle bound on the capillary inner radius (Lemma 3.2); a barrier construction (Lemmas 3.4–3.5) showing $\\max |\\nabla^2 h| \\le C(1+\\|h\\|_{C^0})$, the quantitative estimate that closes the $C^2$ bound; and a geometric lemma (Lemma 2.3) bounding $\\rho_+(\\hat\\Sigma,\\theta)^2/\\rho_-(\\hat\\Sigma,\\theta) \\le C \\max \\lambda_n$, which converts inner-radius control into an outer-radius bound and hence a $C^0$ bound once translations are fixed by capillary evenness. Degree theory on the map $G(h,t) = \\sigma_n(A)/\\sigma_{n-k}(A) - f_t$ then yields existence, with the linearized operator at the constant solution $\\ell$ having kernel exactly the horizontal translations, which the evenness condition kills.","core_discovery":"The central discovery is an existence theorem (Theorem 1.1): for $\\theta \\in (0,\\pi/2]$, $1 \\le k \\le n-1$, and any positive smooth capillary even function $f$ on the spherical cap $C_\\theta$, there is a strictly convex capillary hypersurface $\\Sigma \\subset \\mathbb{R}^{n+1}_+$ with $W_k(\\tilde\\nu^{-1}(\\xi)) = f(\\xi)$ for all $\\xi \\in C_\\theta$. Here $\\tilde\\nu$ is the capillary Gauss map and $W_k$ is the $k$-th elementary symmetric function of the principal curvatures. The problem is equivalent to solving $\\sigma_n(\\nabla^2 h + h\\sigma)/\\sigma_{n-k}(\\nabla^2 h + h\\sigma) = f^{-1}$ on $C_\\theta$ with Robin boundary condition $\\nabla_\\mu h = \\cot\\theta\\, h$, where $h$ is the capillary support function. The main technical work is a set of a priori estimates: a bound on the capillary inner radius, a quantitative linear control of $|\\nabla^2 h|$ by $\\|h\\|_{C^0}$, and a capillary version of a geometric lemma relating the capillary outer and inner radii to the maximal principal radius. These combine into a uniform $C^{4,\\gamma}$ bound that feeds a degree-theoretic existence argument, whose base case is the unique constant-curvature capillary hypersurface. The paper also proves uniqueness for data close to constant (Theorem 1.2) and constructs a one-parameter family of capillary hypersurfaces for which $\\int_{C_\\theta} \\frac{\\langle \\xi, E_\\alpha\\rangle}{W_k}\\, dA_\\sigma \\ne 0$, showing that condition (1.2) is not necessary for $k < n$.","pith_inferences":["The proof uses only that capillary evenness kills the kernel of the linearized operator at the constant solution; any reflection or finite symmetry group of $C_\\theta$ with no nonzero invariant first harmonics would plausibly work the same way, so the symmetry assumption may be more convenient than essential.","The authors expect the theorem to hold for $\\theta>\\pi/2$; if the geometric radius-ratio bound and the boundary barriers can be extended to obtuse contact angles, the identical degree argument would go through, which is a clean test of the method.","The quantitative a priori estimates suggest a numerical continuation from the constant solution $h=\\ell$ along a path of capillary even functions $f$, with uniform step control; the existence could then be exhibited computationally for concrete data.","If evenness is removed, the natural next question is whether a suitably normalized version of condition (1.2) becomes sufficient; the counterexample in Theorem 1.3 shows the raw condition is not necessary, so a sharper obstruction would be needed."],"forward_implications":["For any positive smooth capillary even $f$, the prescribed curvature equation (1.1) has a solution, so the full range $1\\le k\\le n-1$ of Weingarten curvatures is now covered in the capillary setting, not only $k=n$ (the Minkowski case).","At $\\theta = \\pi/2$, reflecting the solution across the boundary plane gives an alternative proof of the classical prescribed Weingarten curvature problem for closed convex hypersurfaces with even symmetry.","The a priori estimate of Theorem 3.1 is quantitative: the $C^{4,\\gamma}$ norm of the support function is controlled by $n$, $k$, $\\gamma$, $\\min f$, and $\\|f\\|_{C^3}$ alone, so the solution family is compact over bounded sets of data.","Near constant data the solution is unique (Theorem 1.2), so there is a well-posed branch of solutions emanating from the unique constant-curvature capillary cap.","The integral condition (1.2), necessary and sufficient for the capillary Minkowski problem, is not necessary for $1\\le k\\le n-1$: the constructed one-parameter family of capillary hypersurfaces violates it."],"supporting_citations":[{"why":"Establishes the capillary support function setup and the equivalence to the Robin boundary problem, and treats the predecessor case k=n whose estimates are refined here.","marker":"[40]"},{"why":"Supplies the Hessian quotient formulation, the degree-theoretic strategy for existence, and the counterexample template used for Theorem 1.3.","marker":"[19]"},{"why":"Source of the geometric lemma bounding the outer/inner radius ratio by the maximal principal radius, adapted as Lemma 2.3.","marker":"[11]"},{"why":"Provides the uniqueness of constant-curvature capillary hypersurfaces (Corollary 1.2) used to fix the degree at the base solution f≡1.","marker":"[28]"},{"why":"Provides the barrier construction that reduces the global C² estimate to a boundary double-normal estimate for Neumann problems, used in Lemmas 3.4–3.5.","marker":"[35]"},{"why":"Supplies the degree theory for second-order nonlinear elliptic operators used to conclude existence from the a priori estimates.","marker":"[31]"},{"why":"Gives the oblique-boundary regularity theory that upgrades the C² estimates to the C^{4,γ} bound of Theorem 3.1.","marker":"[34]"},{"why":"Defines capillary inner and outer radii and the inequalities used to pass from Euclidean to capillary radius bounds.","marker":"[51]"}],"fun_headline_variants":["Capillary surfaces meet prescribed Weingarten curvature","Every symmetric data on a cap yields a convex capillary surface","Solving the capillary prescribed curvature problem in half-space","Convex capillary hypersurfaces: prescribed curvature realized","New estimates settle prescribed curvature for convex capillary surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the capillary evenness of the prescribed function — the symmetry that pins down horizontal translations, without which the a priori bounds driving the proof can fail.","fun_headline_variants_meta":{"raw":{"variants":["Capillary surfaces meet prescribed Weingarten curvature","Every symmetric data on a cap yields a convex capillary surface","Solving the capillary prescribed curvature problem in half-space","Convex capillary hypersurfaces: prescribed curvature realized","New estimates settle prescribed curvature for convex capillary surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001107,"raw_usage":{"total_tokens":4657,"prompt_tokens":1031,"completion_tokens":3626,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":647,"completion_tokens_details":{"reasoning_tokens":3552}},"tokens_in":647,"tokens_out":3626,"duration_ms":24841,"temperature":1.0,"reasoning_tokens":3552,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:54:35.660705+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct numerical or analytical continuation from $f\\equiv1$ along a path of capillary even functions, monitoring the eigenvalues of $\\nabla^2 h + h\\sigma$; if a smooth positive even $f$ with bounded $C^3$ norm produced a support function leaving a fixed $C^{4,\\gamma}$ ball or losing convexity, the a priori estimate of Theorem 3.1 and the degree-theoretic existence proof would be false.","supporting_citations":[{"cited_title":"The capillary Minkowski pr oblem","cited_arxiv_id":null,"evidence_quote":"Establishes the capillary support function setup and the equivalence to the Robin boundary problem, and treats the predecessor case k=n whose estimates are refined here."},{"cited_title":"Convex hypersurfaces of prescribe d curvatures","cited_arxiv_id":null,"evidence_quote":"Supplies the Hessian quotient formulation, the degree-theoretic strategy for existence, and the counterexample template used for Theorem 1.3."},{"cited_title":"A logarithmic Gauss curvatu re flow and the Minkowski problem","cited_arxiv_id":null,"evidence_quote":"Source of the geometric lemma bounding the outer/inner radius ratio by the maximal principal radius, adapted as Lemma 2.3."},{"cited_title":"The Neu mann problem for equa- tions of Monge-Ampère type","cited_arxiv_id":null,"evidence_quote":"Provides the barrier construction that reduces the global C² estimate to a boundary double-normal estimate for Neumann problems, used in Lemmas 3.4–3.5."},{"cited_title":"Degree theory for second order nonlinear ellipt ic operators and its applica- tions","cited_arxiv_id":null,"evidence_quote":"Supplies the degree theory for second-order nonlinear elliptic operators used to conclude existence from the a priori estimates."},{"cited_title":"Nonlinear oblique boundary value problems for nonlinear elliptic equations","cited_arxiv_id":null,"evidence_quote":"Gives the oblique-boundary regularity theory that upgrades the C² estimates to the C^{4,γ} bound of Theorem 3.1."},{"cited_title":"Hypersurfaces with capilla ry boundary evolving by volume preserving power mean curvature flow","cited_arxiv_id":null,"evidence_quote":"Defines capillary inner and outer radii and the inequalities used to pass from Euclidean to capillary radius bounds."}],"review_version":1}