{"id":"9c5146ad-3f5a-416f-9740-c08110bb1f6c","arxiv_id":"2506.02146","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Renormalized capillary area density converges to the Weiss energy, giving angle-independent curvature estimates and a Bernstein theorem for capillary minimizers.","lead":"This paper shows that when the contact angle of a capillary surface tends to zero, a properly rescaled version of the surface's area density converges to the Weiss energy, a monotone quantity from the one-phase free-boundary problem. This convergence yields curvature bounds for capillary minimizers that are independent of the angle, and a Bernstein-type classification of global minimizers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Case 1a of Thm 1.2 relies on [1, Prop 4.11] giving C^{2,α} convergence at points approaching the free boundary; if that proposition only gives convergence away from the interface, the curvature contradiction is incomplete.","rationale":"The paper is a rigorous continuation of the authors' program, and Theorem 1.1's proof is internally consistent: the expansion of the regularized area density, the comparison inequalities (6)–(7), and the limiting argument using continuity of W_v in r all check out. The reader's weakest_assumption points to the dependence on [1] and [2], and that is indeed where the most load-bearing step lives. In Theorem 1.2, Case 1a, the normalized curvature at a point whose scaled distance to the free boundary stays bounded is asserted to pass to |D^2v(0)| via [1, Prop. 4.11]. This is delicate because the limiting AC minimizer is not twice differentiable across the free boundary, so the required convergence must be in a surface-geometric or hodograph sense rather than ordinary C^{2,α} convergence of the graph functions. The paper states the conclusion without specifying which variant of the proposition applies. I do not regard this as a demonstrated error; it is a concrete verification point. Since the curvature estimate and its corollaries depend on it, the cleanest verdict is conditional acceptance pending that check. The reader's confidence was already moderate for related reasons, so this is a refinement rather than a rejection.","tokens_in":9567,"tokens_out":37846,"duration_ms":390653,"concrete_test":"Read the statement and proof of [1, Prop. 4.11]. Verify that for sequences satisfying the Case 1a hypotheses (uniform Lipschitz bound, sup_i θ_i^{-1}|A_{M'_i}|≤4, free boundaries at bounded distance, and the scaled density bound) it yields convergence of θ_i^{-1} times the second fundamental form of M'_i at points with bounded scaled distance to ∂M'_i to the second fundamental form of the limiting interface, not merely C^{2,α} convergence of the normalized graph functions on compact subsets of {v>0} ∪ {v=0}. If the proposition only gives the latter, re-run the contradiction at a point x'_i with dist(x'_i,∂M'_i)≥η>0 and check whether a positive lower bound for λ_i can still be obtained; if not, the small-angle curvature estimate is unproved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.2's small-angle argument (Case 1a) is the crucial step converting the new density convergence into the angle-independent curvature estimate. After rescaling, the chosen point x'_i satisfies θ_i^{-1}|A_{M'_i}(x'_i)|=1 and has bounded scaled distance to ∂M'_i, so the projected point 0 can converge to ∂{v>0}. The proof invokes [1, Prop. 4.11] to obtain θ_i^{-1}u'_i→v in C^{2,α}_{loc}(R^n) and then asserts 1=θ_i^{-1}|A_{M'_i}(x'_i)|→|D^2v(0)|. But a one-phase minimizer v=(x·n)_+ is not C^2 at the free boundary, so ordinary C^{2,α} convergence of the graph functions cannot hold up to ∂{v>0}. What is needed is convergence of the second fundamental form of the scaled capillary surfaces, in hodograph coordinates if necessary, including at points that approach the interface. The paper does not specify which form of [1, Prop. 4.11] is being used here, and Corollary 1.7's caveat about Hodograph transforms suggests the ordinary C^{2,α} statement is only away from the free boundary. If the proposition does not cover boundary-approaching points, the contradiction in Case 1a fails and the curvature estimate does not follow. This is a verification concern about a published dependency, not an observed falsehood; the rest of the proof is coherent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper continues the authors' study of the small-angle limit of minimizing capillary hypersurfaces in a half-space. The main result, Theorem 1.1, states that for a sequence of capillary minimizers with contact angle θ_i → 0, the renormalized capillary area density θ_i^{-2}Θ_{V_i}(x_i,r_i) converges to (1/(2ω_n)) W_v(x,r), the Weiss energy density of the limiting Alt–Caffarelli minimizer. The authors use this convergence to prove an a priori curvature estimate (Theorem 1.2) under the assumption that the capillary density is uniformly close to the flat half-plane value, with constants depending only on the dimension. They then derive a Bernstein-type classification of global minimizers with near-minimal density (Corollary 1.6) and an improved regularity statement for the graphical convergence, formulated via Hodograph transforms near the free boundary (Corollary 1.7). The proofs combine a regularized density expansion with compactness and improved-regularity results from the authors' prior work [1] and related capillary regularity theory [2,3].","tokens_in":9888,"tokens_out":16552,"duration_ms":172034,"significance":"If the main theorems are correct, the paper provides a sharp quantitative bridge between capillary variational problems and the one-phase Bernoulli problem: the monotone quantity of the capillary problem converges to the Weiss energy, and one obtains dimension-dependent curvature bounds that remain uniform as the contact angle tends to zero. The explicit dimension dependence in Theorem 1.2 is a notable strength, and the monotone-quantity convergence in Theorem 1.1 is a natural and potentially widely useful result. The arguments are largely coherent, and the paper leans on external benchmarks rather than ad hoc assumptions. The main caveat, discussed below, is the precise sense in which the improved convergence from [1, Proposition 4.11] is used in the proof of Theorem 1.2; this is a localized but load-bearing point that needs to be clarified or repaired.","major_comments":[{"comment":"The proof invokes [1, Proposition 4.11] to obtain θ_i^{-1}u'_i → v in C^{2,α}_{loc}(ℝ^n) and then uses the normalization (14) to assert 1 = θ_i^{-1}|A_{M'_i}(x'_i)| → |D^2v(0)|. As written, this step is not justified. The limiting function v is a one-phase minimizer with a free boundary, and the rescaling point 0 may lie on ∂{v>0}; in the model case v(y)=(y·n)_+, the function is not C^1 at the free boundary, so ordinary C^{2,α} convergence of the graph functions cannot hold in a neighborhood of such a point. If [1, Proposition 4.11] only provides C^{2,α} convergence away from the free boundary, then the points x'_i, which are allowed to approach the interface, are not covered and the curvature contradiction in Case 1a does not follow. The authors should either quote a version of [1, Proposition 4.11] that gives convergence of the surfaces at boundary-approaching points (for instance, in the Hodograph sense mentioned in Corollary 1.7), or rewrite the normalization step so that the limit of the second fundamental form is obtained from ambient convergence of the hypersurfaces rather than from the Hessian of the graph function at a potentially nonsmooth point. This is a load-bearing gap in the small-angle curvature estimate.","section":"Section 2, Case 1a of Theorem 1.2 (around Eq. (14))"}],"minor_comments":[{"comment":"There is a typo: 'satsfying' should be 'satisfying'.","section":"Equation (1)"},{"comment":"'The a salient aspect' should be 'A salient aspect'.","section":"Remark 1.3"},{"comment":"In the display after (8), 'similary' should be 'similarly'.","section":"Proof of Theorem 1.1"},{"comment":"The phrase 'any xi ∈ ∂ℝ^{n+1}_+ → x ∈ B^n_1' is not a well-formed convergence statement; it should say 'any sequence xi → x with xi ∈ ∂ℝ^{n+1}_+'.","section":"Theorem 1.1 statement"},{"comment":"The statement that the convergence θ_i^{-1}u_i → v is 'in fact C^{2,α}_{loc}(B_1)' should be qualified as in Corollary 1.7, since for a one-phase minimizer with a free boundary this cannot hold in the usual sense at boundary points.","section":"Introduction, after (1)"},{"comment":"'If Ω≠ ∅ or ℝ^{2}_+' should presumably be 'If Ω≠ ∅ and Ω≠ ℝ^{2}_+'; otherwise the sentence is ambiguous.","section":"Lemma 2.1, n = 1 case"},{"comment":"'Defined the rescaled domains' should be 'Define the rescaled domains'.","section":"Case 2a of Theorem 1.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong continuation of the authors' program, and the monotone-quantity convergence in Theorem 1.1 appears sound. The main issue is localized to Case 1a of Theorem 1.2, where the proof relies on a version of [1, Proposition 4.11] whose precise content is not stated. I believe this is repairable if the authors make explicit the Hodograph/ambient convergence notion and adapt the curvature-limit step accordingly. However, because Theorem 1.2 is a central claim, the manuscript should not be accepted until this point is resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result here is Theorem 1.1: for capillary minimizers with contact angle θ_i → 0, the renormalized density θ_i^{-2} Θ_{V_i}(x_i, r_i) converges to (1/(2ω_n)) W_v(x, r), the Weiss energy density of the limiting Alt–Caffarelli minimizer. That is a clean, new statement, and the proof is careful. The regularized density expansion is handled with explicit error terms, and the limiting argument via the known Lipschitz bound and the monotonicity of both quantities is sound. This gives a real bridge between capillary monotonicity and one-phase free-boundary monotonicity, and it is a nice step beyond the authors' earlier convergence results.\n\nThe applications are exactly what you would hope for: angle-independent curvature estimates (Theorem 1.2), a Bernstein-type theorem (Corollary 1.6), and C^{2,\\alpha} regularity in the sense of Hodograph transforms (Corollary 1.7). The curvature estimate, with constants depending only on dimension, would be a major advance if it fully holds. The proof of Theorem 1.2 is a standard contradiction–compactness argument, and most of it reads well. The cone classification lemma is routine but usefully reproduced.\n\nNow the soft spot. In Case 1a of Theorem 1.2, after rescaling, the point x'_i has bounded distance to the free boundary, so the projected point can land on ∂{v>0} in the limit. The proof asserts convergence of θ_i^{-1}|A_{M'_i}(x'_i)| to |D^2v(0)|, using the improved convergence of [1, Prop 4.11]. But if v is the half-plane solution (y·n)_+, it is not C^2 across the free boundary, so ordinary C^{2,α} convergence of the graph functions cannot hold at boundary-approaching points. The paper does not state whether [1, Prop 4.11] provides convergence of the second fundamental form in the Hodograph sense up to the free boundary, and Corollary 1.7's caveat suggests the ordinary statement is only away from it. If the proposition does not cover boundary-approaching points, the contradiction in Case 1a does not go through as written. This is a verification concern about a published dependency, not an observed falsehood; someone should read the statement of [1, Prop 4.11] carefully before trusting Theorem 1.2.\n\nThe paper is for specialists in geometric measure theory and free-boundary regularity. The convergence theorem is solid and deserves to be known. The curvature estimate is important but currently rests on a subtle external dependence. I would send this to a serious referee, and specifically ask them to check that dependency.\n\nRecommendation: engage with it. The core ideas are good, the main new theorem is real, and the soft spot is a checkable point.","headline":"The paper proves a genuinely new convergence theorem for capillary densities to the Weiss energy and uses it to derive angle-independent curvature estimates; the main proof is coherent, but one step in Theorem 1.2 leans on a published proposition whose boundary-regularity content needs verification.","tokens_in":10415,"tokens_out":7787,"would_cite":true,"duration_ms":82100,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R35","49Q10","53A10","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that capillary surface density, renormalized by the square of the contact angle, converges to the Weiss energy of the limiting one-phase Bernoulli problem, yielding curvature bounds whose constants depend only on the…","keywords":["capillary surfaces","Alt–Caffarelli functional","one-phase Bernoulli problem","Weiss monotonicity formula","density ratio","curvature estimates","Bernstein-type theorem","free boundary regularity"],"falsifier":"Compute the two sides of the limit identity for a family of small-angle capillary minimizers whose limiting Alt–Caffarelli free boundary is a non-flat cone: the theorem predicts that $\\theta_i^{-2}\\Theta_{V_i}(x_i,r_i)$ converges to the cone's Weiss energy divided by $2\\omega_n$, so a discrepancy would refute Theorem 1.1.","tokens_in":9373,"feed_emoji":"🌊","tokens_out":15804,"duration_ms":141376,"temperature":0.7,"pith_summary":"Capillary surfaces are mathematical models of fluid interfaces meeting a container wall at a fixed contact angle $\\theta$. Earlier work of the same authors showed that as $\\theta$ tends to zero, these surfaces can be written as graphs over the container wall and converge, after rescaling by $\\theta$, to minimizers of the Alt–Caffarelli energy, the one-phase Bernoulli problem. This paper establishes that in the same limit the capillary area-density ratio, renormalized by $\\theta^{-2}$, converges to the Weiss energy density of the limiting problem. That convergence is then used to prove that if a capillary minimizer's density is everywhere close to the flat half-plane value, the interface curvature is bounded by a constant times $\\sin\\theta$, with the constant depending on the dimension alone. Consequently, global minimizers satisfying the same near-minimal density condition must be capillary half-planes.","feed_headline":"Capillary density converges to Weiss energy as angle shrinks","feed_subtitle":"This yields curvature bounds and a Bernstein classification with constants that depend only on dimension.","key_machinery":"The load-bearing object is the pair of monotone quantities and their regularized versions. For the capillary problem, the varifold $V = [\\partial\\Omega\\cap \\mathbb{R}^{n+1}_+] - \\cos\\theta\\,[\\partial\\Omega\\cap \\partial\\mathbb{R}^{n+1}_+]$ has a density ratio $\\Theta_V(x,r) = \\|V\\|(B_r(x))/(\\omega_n r^n)$, monotone in $r$ for boundary-plane points. For the Alt–Caffarelli functional, the Weiss energy $W_v(x,r) = r^{-n}\\int_{\\{v>0\\}\\cap B_r(x)}(|Dv|^2+1)\\,dy - r^{-n-1}\\int_{\\partial B_r(x)} v^2\\,d\\sigma$ is monotone in $r$ for stationary solutions. A smooth cutoff $\\zeta$ regularizes both quantities without changing their asymptotics up to the factor $(1-\\varepsilon)^n$. The key computation expresses $\\omega_n r_i^n\\Theta^\\zeta_{V_i}(x_i,r_i)$ as $\\theta_i^2/2$ times the regularized Weiss integrand in the graphical coordinates $u_i$, plus controlled $O(\\theta_i)$ errors; renormalizing by $\\theta_i^{-2}$ and taking the limit produces the identity of Theorem 1.1.","core_discovery":"The central discovery is a limit identity between the monotone quantities of two variational problems. Let $\\Omega_i$ be smooth minimizers of the capillary functional $A_{\\theta_i}(\\Omega) = \\mathcal{H}^n(\\partial^*\\Omega\\cap \\mathbb{R}^{n+1}_+) - \\cos\\theta_i\\, \\mathcal{H}^n(\\partial^*\\Omega\\cap \\partial\\mathbb{R}^{n+1}_+)$ with $\\theta_i\\to 0$, and let $V_i = [\\partial\\Omega_i\\cap \\mathbb{R}^{n+1}_+] - \\cos\\theta_i\\,[\\partial\\Omega_i\\cap \\partial\\mathbb{R}^{n+1}_+]$ be the associated capillary varifolds. If $v$ is the limiting Alt–Caffarelli minimizer obtained by the authors' earlier convergence theorem, Theorem 1.1 asserts that $\\theta_i^{-2}\\Theta_{V_i}(x_i,r_i)\\to \\frac{1}{2\\omega_n} W_v(x,r)$ for $x_i\\to x$ in the boundary plane and $r_i\\to r$, where $\\Theta_{V_i}$ is the density ratio and $W_v$ is the Weiss energy. The proof approximates both quantities by cutoff versions, expands the capillary area in the graphical coordinates $u_i$ using $\\operatorname{Lip}(u_i)\\le c(n)\\theta_i$, and passes to the limit. This identity is the engine for the paper's applications: a priori curvature bounds $|A_M(x)|\\le c(n)\\sin\\theta$ under a density-closeness hypothesis, and the Bernstein-type classification of global minimizers with near-minimal density as capillary half-planes.","pith_inferences":["The same renormalized convergence is likely to hold for complements, with $\\theta$ replaced by $\\pi-\\theta$, following the orientation symmetry noted in Remark 1.4; testing this on explicit capillary half-planes with obtuse angles would be a direct check.","Because the proof uses only the Lipschitz bound and Hausdorff convergence of free boundaries, the identity may extend to sequences of almost-minimizers or to weighted variants of the Alt–Caffarelli functional; the paper does not assert this.","A quantitative refinement seems available: for regular limiting free boundaries, Weiss monotonicity could bound the rate of convergence in the identity, upgrading the curvature estimate to an explicit radius-dependent bound; the paper does not pursue rates.","The density-closeness hypothesis is an $L^\\infty$-type condition on a monotone quantity, so a natural testable question is whether the curvature bound survives under an averaged or integral version of the condition, since monotonicity might enforce pointwise closeness at nearby scales."],"forward_implications":["If a capillary minimizer's density ratio is at every boundary point and radius at most $(1+\\varepsilon)(1-\\cos\\theta)/2$ (up to the negative part of $\\cos\\theta$), then the curvature of its interface is bounded by $c(n)\\sin\\theta$ near the boundary, with constants independent of the angle.","A global smooth capillary minimizer satisfying the same near-minimal density condition is necessarily a capillary half-plane: the scale-invariant curvature bound forces the second fundamental form to vanish after rescaling.","When the limiting Alt–Caffarelli minimizer $v$ is regular at a free-boundary point, the rescaled capillary surfaces converge to $v$ in $C^{2,\\alpha}$ near that point, interpreted through Hodograph transforms.","In the small-angle blow-up argument, density closeness combined with the Weiss monotonicity formula forces the limiting Alt–Caffarelli minimizer to be linear, which turns the curvature blow-up assumption into a contradiction."],"supporting_citations":[{"why":"Supplies the graphical representation with Lipschitz constant of order θ_i, the convergence to the Alt–Caffarelli minimizer, and the improved convergence theorem used throughout.","marker":"[1]"},{"why":"Supplies the compactness, monotonicity, and cone-classification tools for capillary minimizers used in Lemma 2.1 and in the contradiction arguments.","marker":"[2]"},{"why":"Establishes the stationary capillary varifold framework, its monotonicity formula, and the compactness and boundary-regularity results for capillary minimizers.","marker":"[3]"},{"why":"Provides the continuity of the Weiss energy in radius used to send the regularization parameter to zero in the proof of Theorem 1.1.","marker":"[5]"},{"why":"Provides the Weiss monotonicity formula and the value ω_n/2 of the Weiss energy at regular free-boundary points.","marker":"[6]"}],"fun_headline_variants":["Angle-zero limit turns capillary density into Weiss energy","Capillary density converges to Weiss energy at zero angle","Zero-angle limit: capillary density becomes Weiss energy","Weiss energy emerges from capillary density at zero angle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main convergence and curvature estimates depend on the prior regularity result that, for small contact angles, the capillary interface can be written as a function over the container wall whose slope is bounded by a constant times $\\theta_i$; if that bound failed or had a different order in $\\theta_i$, the renormalized density would not converge to the Weiss energy.","fun_headline_variants_meta":{"raw":{"variants":["Angle-zero limit turns capillary density into Weiss energy","Capillary density converges to Weiss energy at zero angle","Zero-angle limit: capillary density becomes Weiss energy","Weiss energy emerges from capillary density at zero angle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001292,"raw_usage":{"total_tokens":5261,"prompt_tokens":917,"completion_tokens":4344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":4283}},"tokens_in":533,"tokens_out":4344,"duration_ms":30389,"temperature":1.0,"reasoning_tokens":4283,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:29:59.912152+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two sides of the limit identity for a family of small-angle capillary minimizers whose limiting Alt–Caffarelli free boundary is a non-flat cone: the theorem predicts that $\\theta_i^{-2}\\Theta_{V_i}(x_i,r_i)$ converges to the cone's Weiss energy divided by $2\\omega_n$, so a discrepancy would refute Theorem 1.1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the graphical representation with Lipschitz constant of order θ_i, the convergence to the Alt–Caffarelli minimizer, and the improved convergence theorem used throughout."},{"cited_title":"Regularity of minimal surfaces with capillary boundary conditions","cited_arxiv_id":"2405.20796","evidence_quote":"Supplies the compactness, monotonicity, and cone-classification tools for capillary minimizers used in Lemma 2.1 and in the contradiction arguments."},{"cited_title":"De Philippis and F","cited_arxiv_id":null,"evidence_quote":"Establishes the stationary capillary varifold framework, its monotonicity formula, and the compactness and boundary-regularity results for capillary minimizers."},{"cited_title":"28, Springer, Cham, [2023] ©2023","cited_arxiv_id":null,"evidence_quote":"Provides the continuity of the Weiss energy in radius used to send the regularization parameter to zero in the proof of Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Weiss monotonicity formula and the value ω_n/2 of the Weiss energy at regular free-boundary points."}],"review_version":1}