{"id":"28d0b828-b7d6-40af-9328-54d87fcdaf07","arxiv_id":"2502.07697","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Minimizing capillary cones are flat in dimension 4 when the free-boundary mean curvature has one sign, and axially symmetric ones are flat up to dimension 6.","lead":"The paper proves that minimizing capillary cones in dimension 4 with non-sign-changing free-boundary mean curvature are flat, and that axially symmetric minimizing cones are flat in dimensions up to 6. If correct, it sharpens the known bound on the singular set of capillary free boundaries from codimension 4 to codimension 5 for graphical minimizers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The strict interior improvement in Theorem 1.1, Case 2 is asserted but not proved; with Λ=4/9 exactly at the borderline, Proposition 2.4 requires a strictness that the sketched argument does not establish.","rationale":"The reader's identified weakness is exactly the load-bearing point. The theorem depends on a strict inequality at the borderline constant 4/9, and the proof's only source of strictness in Case 2 is the asserted improvement of (4.7) near the K2-attaining points. The text gives no detailed derivation of the uniform δ, no demonstration that the improved c-level estimate survives passage to w = c^{1/3}, and no verification that the strictness holds in the distributional sense needed to apply Proposition 2.4. These are not cosmetic omissions: any failure of strictness leaves the stability criterion inapplicable. The gap appears fillable by a direct algebraic and elliptic estimate, which is why the appropriate verdict remains CONDITIONAL rather than REJECT; the reader's conditional assessment is unchanged. Other potential issues, such as the regularity of the piecewise-quadratic function f in (4.10) and the handling of the H∂M = 0 subset, are secondary: they are either standard regularization matters explicitly flagged in Remark 3.2 or can be absorbed into the same strictness check. The central claim is plausible, but the paper should supply the omitted proof step before the result is relied upon.","tokens_in":27841,"tokens_out":29826,"duration_ms":239684,"concrete_test":"Independently re-derive Case 2 of Theorem 1.1 at a point where K2 is attained. Using the coordinate system of §2.1 and the competitor (4.13), compute explicitly the factor S4 := Σ_{i≠{1,4}} (λi − λ4)(f_{λi} − f_{λ4}) appearing in (4.7) as a function of the eigenvalue ratios near the configuration (λ1,λ2,λ3,λ4) = (0, −2λ4, λ4, λ4). Verify that there is a neighborhood on which S4/c ≤ 3/2 + δ < 3 for a uniform δ > 0, and then check that this strict gain improves (4.1) for c and, after the ε-regularization in Lemma 4.2, yields a strict version of (4.14) for w = c^{1/3} with coefficient strictly greater than 4/9. If the strict coefficient cannot be established, Proposition 2.4 does not apply and the proof of Theorem 1.1 is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.1, Case 2 (H∂M ≥ 0), the boundary inequality is an equality on the set where the algebraic maximum K2 = 3 in (4.17) is attained, corresponding to λ1 = 0, λ2 = −2λ4, λ3 = λ4. The interior inequality (4.14) has the borderline constant Λ = 4/9, exactly equal to (n/2 + k − 1)^2 in (4.19), so Proposition 2.4 can be applied only if the interior inequality is strict. To obtain this strictness, the authors assert that, since λ2 ≠ λ3 and λ4 ≠ 0 in a neighborhood, inequality (4.7) for k = 4 can be improved to c4² ≤ (3/2 + δ)(Σ ∂²_{ai4}F(A) u²_{ii4}) c < 3(Σ ...) c in N_{x0}, and that 'continuing the proof as in Lemma 4.1' yields a strict interior inequality. This is the load-bearing step. It is only sketched: the continuity argument producing a uniform δ > 0 is not given, the propagation of the improved estimate for c to the α = 1/3 power w = c^{1/3} through the ε-regularization in Lemma 4.2 is not shown, and there is no explicit check that the strictness survives in the distributional sense required by Proposition 2.4. If the improvement is only non-strict, or if the regularization erases the gain, the application of Proposition 2.4 fails and Theorem 1.1 is unproved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimizing capillary cones with free boundary, i.e. homogeneous minimizers of the capillary energy in a half-space. The main theorem (Theorem 1.1) asserts that in dimension n=4, a minimizing capillary cone with contact angle θ∈(0,π) is flat whenever the mean curvature H∂M of the free boundary has constant sign. As applications, the authors deduce that graphical capillary minimizers have free boundary singular set of Hausdorff dimension at most n−5 (Corollary 1.4), and they prove instability of non-trivial axially symmetric capillary cones in dimensions up to 6 (Theorem 1.2). The technical core is a stability criterion à la Jerison–Savin (Proposition 2.4), a generalized Simons-type inequality for convex homogeneous symmetric functions of the second fundamental form (Lemmas 4.1 and 4.2), and a boundary identity for a carefully chosen quadratic competitor (Lemma 4.3 and Section 4.3).","tokens_in":28183,"tokens_out":24659,"duration_ms":200585,"significance":"If the main results are correct, they give the first contact-angle-independent improvement of the free boundary singularity dimension for capillary hypersurfaces, in the graphical case exceeding the previously known threshold in dimension 4. The paper is methodologically valuable: it transfers the Jerison–Savin one-phase strategy to capillary cones, introduces an explicit quadratic competitor that is claimed to be optimal in a natural class, and provides self-contained boundary identities. The derivations are not circular: the rigidity conclusions follow from stability inequalities and geometric competitors, with no parameter fitted to the conclusion. However, the proof of the central theorem currently contains a sign error in Case 1 and an unproved strict-improvement assertion in Case 2; these issues are load-bearing and must be resolved before the results can be accepted.","major_comments":[{"comment":"The boundary inequality for the case H∂M≤0 is not established by the displayed argument. The text states that H∂M(1−αL) ≥ H∂M/2 > 0 on {H∂M<0}, but when H∂M<0 the right-hand side is negative, so this inequality does not imply the required boundary inequality H∂M(1−αL)≥0. In fact, at a free-boundary point with λ1=0 and λ2=λ3=−λ4/2, λ4>0 (which corresponds to H∂M<0), one computes L=3/2, and with α=1/3 one has 1−αL=1/2, so H∂M(1−αL)<0. Thus the chosen α=1/3 is not admissible for H∂M<0. A correct treatment of Case 1 should choose α=2/3 instead, since then the boundary condition holds (because L≥3/2 gives 1−αL≤0) and the interior inequality gives Λ=10/9, which is strictly larger than (n/2−α−1)^2=1/9, so Proposition 2.4 applies without needing strictness. The authors should correct this sign error and revise the proof accordingly.","section":"§4.3, Case 1"},{"comment":"The strict interior improvement that is needed to apply Proposition 2.4 is asserted but not proved. The paragraph after (4.19) claims that because λ2≠λ3 and λ4≠0 at points where the boundary condition is an equality, there exists δ>0 such that c4² ≤ (3/2+δ)(Σ_{i≠{1,4}} ∂²_{a_i4}F(A)u²_{ii4}) c in a neighborhood, and that 'continuing the proof as in Lemma 4.1' yields a strict interior inequality. This is the load-bearing step: with Λ=4/9 exactly equal to the Hardy constant in (4.19), Proposition 2.4 can be applied only if the inequality is strict. The manuscript does not provide the continuity argument producing a uniform δ>0, does not show how the improved estimate for c propagates to w=c^{1/3} through the ε-regularization in Lemma 4.2, and does not check that strictness survives in the distributional sense required by Proposition 2.4. The authors must supply a complete proof of this strict improvement, or replace it with another argument that yields the strictness.","section":"§4.3, Case 2"},{"comment":"Lemma 4.2, which is central to both Theorem 1.1 and Theorem 1.2, states that its proof is omitted and refers to 'repeating the same computations as in the proof of Lemma 3.1'. This is not adequate for a lemma whose strict version is used in the borderline Case 2 of Theorem 1.1. The regularization argument for a general convex homogeneous f, the behavior of the strict inequality under the limit ε→0+, and the precise hypotheses on f (for example, whether the f in (4.10) is strictly convex) need to be written out. In particular, the strictness assertion in the last sentence of Lemma 4.2 requires a proof; it is not a formal consequence of the displayed computation in Lemma 3.1 without additional assumptions on the nodal set of c.","section":"Lemma 4.2"},{"comment":"In the proof of Theorem 1.2 for n=6, the assertion that (∇_M(log|A|)·η)^2>0 on ∂M is stated without justification. This strictness is what upgrades the interior inequality from the critical constant to a strict one in the borderline dimension. The claim is plausible and can be derived from Lemma 3.3 together with the axial symmetry relations λ_j=−λ_n/(n−2); however, the derivation is not given. Since this step is needed for the n=6 case of Theorem 1.2, the authors should include the computation.","section":"Theorem 1.2, n=6 case"}],"minor_comments":[{"comment":"The notation in the proof is confusing: the test function is first called φ and then ϕ:=wφ is used with the same letter, and the boundary term contains (∇_M w²·φ) where it should presumably be (∇_M w²·η). Please clarify the notation and correct the typo.","section":"§2.2, proof of Proposition 2.4"},{"comment":"The phrase 'all the principal curvatures are equal' is imprecise, because the radial principal curvature of ∂M is zero; the intended statement is that the n−2 non-radial principal curvatures are equal. Please rephrase.","section":"§3.3, Theorem 1.2"},{"comment":"For n=2, the chosen exponent α=(n−2)/(n−1) is 0, which is outside the admissible range α∈(0,1) used in Lemma 3.4. If Theorem 1.2 is intended to include n=2, this case should be treated separately; otherwise the statement should explicitly assume n≥3.","section":"§3.3, Theorem 1.2"},{"comment":"In the proof, the notation ∂_{a_{ij}}²F²(A) is used before F is defined, and the line '∂_{a_{ij}}²F²(A)=...' would be clearer if written as 2∂_{a_{ij}}²[F(A)²] or an explicitly defined symbol. This is a readability issue only.","section":"§4.1, Lemma 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is squarely within the scope of the journal and addresses a significant problem. The main concern is not novelty or circularity but rigor in the proof of Theorem 1.1. The sign error in Case 1 is concrete and must be fixed; it is likely repairable by choosing α=2/3 for that case. The strict-improvement argument in Case 2 is the true heart of the paper and needs a complete proof; if it cannot be supplied, Theorem 1.1 would remain unproved. The omission of the proof of Lemma 4.2 should be addressed regardless. I recommend major revision rather than rejection because the claimed results appear credible and the identified gaps are localized and plausibly fixable within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a real contribution to capillary free-boundary regularity, and the Jerison–Savin strategy transplants cleanly into the capillary setting. The new results are Theorem 1.1 (flatness in n=4 under a sign condition on H∂M), Theorem 1.2 (axial symmetry forces flatness for n<7), and the graphical corollary n*_G≥5. None of these is in [7] or [22], and the n=4 quadratic competitor with a=4 in (4.13) is genuinely new. The boundary identities in Lemmas 2.1, 3.3, and 4.3 are worked out in real detail, and Lemma 4.1 is a useful generalized Simons inequality with a strictness observation that matters. The paper is also honest about the connection to the one-phase Bernoulli problem. No circularity: the rigidity is deduced from stability, not fitted to the conclusion.\n\nThe soft spots are concentrated in the proof of Theorem 1.1, Case 2. Lemma 4.2 is not proved—the proof is omitted—and while the ε-regularization is standard, it is exactly the step that must preserve strictness. Proposition 2.4 is only sketched, and its distributional-strictness hypothesis needs care. The load-bearing issue is the asserted strict interior improvement near points where the boundary inequality is an equality. The constant in (4.14) is exactly borderline: 4/9 equals (n/2+k−1)^2, so Proposition 2.4 applies only with strictness. The text says that λ2≠λ3 and λ4≠0 give δ>0 and an improved estimate for c4^2, then “continuing the proof as in Lemma 4.1” yields a strict interior inequality. That continuity/uniformity argument and its propagation through the ε-regularization are not shown. If the improvement is only non-strict, or if the regularization erases it, the proof of Theorem 1.1 does not go through. This is not a manufactured objection; it is the exact point where the borderline constant leaves no slack.\n\nVerdict: the architecture is right and the results are probably true, but the manuscript currently has a load-bearing gap in Case 2 plus two omitted technical proofs. A serious referee could get the paper into shape, but the gap should not be waved through. I would send it to review, and I would bring it to reading group—the technique is worth understanding even as the strictness point is being resolved.","headline":"A serious contribution to capillary free-boundary regularity with a load-bearing strictness gap in the proof of Theorem 1.1, Case 2.","tokens_in":28740,"tokens_out":2092,"would_cite":true,"duration_ms":20124,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q05","49Q10","35R35","35B07","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that in dimension n=4, a minimizing capillary cone whose free-boundary mean curvature has constant sign must be flat, and that the singular set of graphical capillary minimizers has Hausdorff dimension at most n−5.","keywords":["capillary surfaces","singularity","global stable solutions","one-phase Bernoulli problem","free boundary regularity","minimizing cones","Simons-type inequality","Hausdorff dimension"],"falsifier":"Perform the omitted computation in Section 4.3, Case 2: at a free-boundary point with $\\lambda_1=0$, $\\lambda_2=-2\\lambda_4$, $\\lambda_3=\\lambda_4$, decide whether $c_4^2<3(\\sum_{i\\neq1,4}\\partial^2_{a_{i4}}F(A)\\,u^2_{ii4})c$ holds with a positive margin; if the inequality is only an equality, Proposition 2.4 cannot be applied. Alternatively, exhibit a non-flat minimizing capillary cone in $\\mathbb{R}^5$ with $H_{\\partial M}$ of one sign: Theorem 1.1 says that none exists.","tokens_in":27635,"feed_emoji":"💧","tokens_out":10907,"duration_ms":98703,"temperature":0.7,"pith_summary":"This paper is trying to show that singularities of minimizing capillary surfaces with free boundary can be ruled out in low dimensions once the mean curvature of the free boundary does not change sign. The main theorem says that in dimension four every minimizing capillary cone with one-signed free-boundary mean curvature is flat. If true, this lowers the expected dimension threshold for the singular set of graphical capillary minimizers to n−5, improving the previous n−4 bound, and it parallels the one-phase Bernoulli problem, where a similar stability criterion already forces rigidity. The paper also establishes the instability of non-trivial axially symmetric capillary cones in dimensions up to six. A sympathetic reader should care because these cones are the blow-up profiles that govern the regularity of capillary droplets, and the result excludes the low-dimensional singular behavior under a mild geometric sign condition.","feed_headline":"Dimension-4 capillary cones with one-sign curvature are flat","feed_subtitle":"A new stability test bounds the singular set of graphical capillary minimizers by dimension n−5.","key_machinery":"The load-bearing object is a stability criterion (Proposition 2.4): if a $k$-homogeneous nonnegative function $w$, smooth on its positivity set, satisfies an interior inequality $\\frac12\\Delta_M w^2-|\\nabla_M w|^2+|A|^2w^2\\ge\\Lambda|x|^{-2}w^2$ and a boundary inequality $\\cos(\\theta)H_{\\partial M}w^2-\\frac12(\\nabla_M w^2\\cdot\\eta)\\ge0$, with $\\Lambda\\ge(n/2+k-1)^2$, and at least one of these is strict, then $w\\equiv0$. The paper produces such $w$ as a power $w=c^\\alpha$ of a convex, one-homogeneous, symmetric function $c=F(A)$ of the principal curvatures, for which Lemma 4.1 proves a Simons-type inequality. The decisive competitor in dimension four is $w=c^{1/3}$ with $c=(\\sum_{\\lambda_i\\ge0}\\lambda_i^2+4\\sum_{\\lambda_s<0}\\lambda_s^2)^{1/2}$; the coefficient $4$ is chosen so that the boundary quotient $L(x)$ stays on the correct side of $\\alpha=1/3$, and at the extremal free-boundary configuration the argument needs a strict improvement of the curvature estimate (4.7).","core_discovery":"The central claim is Theorem 1.1: if $M$ is a minimizing capillary cone with contact angle $\\theta\\in(0,\\pi)$ and the mean curvature $H_{\\partial M}$ of its free boundary has constant sign on $\\{x_{n+1}=0\\}$, then in dimension $n=4$ the cone is flat, so the singularity at the origin is ruled out. The proof feeds a weighted quadratic function of the principal curvatures into a stability criterion: the competitor $w=c^{1/3}$ with $c=(\\sum_{\\lambda_i\\ge0}\\lambda_i^2+4\\sum_{\\lambda_s<0}\\lambda_s^2)^{1/2}$ satisfies a Simons-type interior inequality and a boundary inequality governed by the sign of $H_{\\partial M}$; when one of the inequalities is strict, the criterion forces $w\\equiv0$, hence the second fundamental form vanishes and the cone is flat. Because graphical capillary cones automatically have $H_{\\partial M}$ of one sign, the same argument gives $n_*^G(\\theta)\\ge5$ and, by Federer's dimension reduction, Hausdorff dimension at most $n-5$ for the free-boundary singular set of graphical capillary minimizers (Corollary 1.4). A separate power-of-$|A|$ argument shows that axially symmetric capillary cones are flat in dimensions $n<7$ (Theorem 1.2).","pith_inferences":["If the strict-improvement step in Section 4.3, Case 2 can be made fully rigorous, the borderline constant suggests that the sign assumption on $H_{\\partial M}$ may be removable in dimension four; searching for non-graphical minimizing cones with changing-sign $H_{\\partial M}$ would test this.","The parallel with the one-phase Bernoulli problem suggests the optimal critical dimension for general capillary cones may be $n=5$, matching the graphical threshold; proving flatness of all minimizing cones in dimension four without the sign condition would settle it.","A concrete extension is to replace the quadratic competitor by higher-degree symmetric functions of the principal curvatures; the extremal-configuration computation in (4.17) would show whether the dimension-four cutoff is an artifact of the quadratic ansatz."],"forward_implications":["For graphical capillary drops, the free-boundary singular set has Hausdorff dimension at most $n-5$, so the free boundary is smooth in dimensions up to four.","In dimensions $n<7$, every minimizing capillary cone with axially symmetric free boundary is flat; non-trivial axially symmetric cones are therefore unstable.","The stability criterion extends the one-phase Bernoulli rigidity picture to capillary cones, giving a unified way to turn curvature inequalities into vanishing of the second fundamental form.","Within the quadratic class of curvature competitors, the coefficient $a=4$ is the only choice that makes the stability argument close in dimension four (Remark 4.5)."],"supporting_citations":[{"why":"Supplies the stability criterion and the one-homogeneous convex-function competitor method that the paper adapts to the capillary setting.","marker":"[22]"},{"why":"Supplies the Simons identity and inequality for the second fundamental form that Lemma 4.1 generalizes.","marker":"[30]"},{"why":"Provides the standard curvature-estimate and radial-test-function argument used to prove the stability criterion.","marker":"[28]"},{"why":"Established the previous free-boundary regularity threshold and the rigidity for contact angles near the critical values, which this paper improves.","marker":"[7]"},{"why":"Provides the one-phase Bernoulli stability inequality that motivates the boundary inequality and the constant-sign mean-curvature fact for graphical solutions.","marker":"[5]"},{"why":"Established the optimal threshold $n=7$ for the perpendicular-contact-angle case, serving as the benchmark for the axially symmetric result.","marker":"[17]"},{"why":"Introduced the variational formulation for graphical capillary drops that makes Corollary 1.4 applicable.","marker":"[4]"}],"fun_headline_variants":["Capillary cones: flatness up to dimension 4 under one-sign curvature","Stability criterion flattens minimizing capillary cones in dims ≤4","Flat capillary cones in dims ≤4 under one-sign mean curvature","Singular capillary cones flat up to n=4 with constant-sign curvature","Instability of axially symmetric capillary cones in dims ≤6"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In the case $H_{\\partial M}\\ge0$, the proof needs a strict quantitative improvement of the Simons-type inequality (4.7) for the component $k=4$ near free-boundary points where the boundary inequality is an equality; the authors state that the improvement follows because $\\lambda_2\\neq\\lambda_3$ and $\\lambda_4\\neq0$, but the computation is not carried out, and the borderline constant $4/9$ leaves no room for a non-strict version.","fun_headline_variants_meta":{"raw":{"variants":["Capillary cones: flatness up to dimension 4 under one-sign curvature","Stability criterion flattens minimizing capillary cones in dims ≤4","Flat capillary cones in dims ≤4 under one-sign mean curvature","Singular capillary cones flat up to n=4 with constant-sign curvature","Instability of axially symmetric capillary cones in dims ≤6"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00077,"raw_usage":{"total_tokens":3404,"prompt_tokens":931,"completion_tokens":2473,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":2378}},"tokens_in":547,"tokens_out":2473,"duration_ms":16826,"temperature":1.0,"reasoning_tokens":2378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:51:42.929509+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform the omitted computation in Section 4.3, Case 2: at a free-boundary point with $\\lambda_1=0$, $\\lambda_2=-2\\lambda_4$, $\\lambda_3=\\lambda_4$, decide whether $c_4^2<3(\\sum_{i\\neq1,4}\\partial^2_{a_{i4}}F(A)\\,u^2_{ii4})c$ holds with a positive margin; if the inequality is only an equality, Proposition 2.4 cannot be applied. Alternatively, exhibit a non-flat minimizing capillary cone in $\\mathbb{R}^5$ with $H_{\\partial M}$ of one sign: Theorem 1.1 says that none exists.","supporting_citations":[{"cited_title":"Jerison, O","cited_arxiv_id":null,"evidence_quote":"Supplies the stability criterion and the one-homogeneous convex-function competitor method that the paper adapts to the capillary setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Simons identity and inequality for the second fundamental form that Lemma 4.1 generalizes."},{"cited_title":"Schoen, L","cited_arxiv_id":null,"evidence_quote":"Provides the standard curvature-estimate and radial-test-function argument used to prove the stability criterion."},{"cited_title":"Caffarelli, D","cited_arxiv_id":null,"evidence_quote":"Provides the one-phase Bernoulli stability inequality that motivates the boundary inequality and the constant-sign mean-curvature fact for graphical solutions."},{"cited_title":"Gr¨ uter.Optimal regularity for codimension one minimal surfaces wi th a free boundary","cited_arxiv_id":null,"evidence_quote":"Established the optimal threshold $n=7$ for the perpendicular-contact-angle case, serving as the benchmark for the axially symmetric result."},{"cited_title":"Caffarelli, A","cited_arxiv_id":null,"evidence_quote":"Introduced the variational formulation for graphical capillary drops that makes Corollary 1.4 applicable."}],"review_version":1}