{"id":"9d7dde7a-6888-46cd-82da-d97c876491ba","arxiv_id":"1908.06397","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The concave minimal-graph solution over an (a,eta)-type convex domain in hyperbolic space is C^{max{1/a,1/(n+1)}} with explicit constants, refining the known C^{1/(n+1)} bound.","lead":"This paper proves that the global Hölder exponent for the solution of the minimal graph Dirichlet problem in hyperbolic space is determined by a geometric boundary quantity, the type a, rather than by boundary smoothness. The result sharpens earlier work by Han, Shen and Wang and provides explicit estimates with constants depending on a, eta, n and the domain diameter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The coordinate-alignment step in Theorems 2.6 and 3.1 is the weakest point: the proof needs a lemma that nearest boundary points of interior points are regular, otherwise the (a,eta)-type containment may not hold in the rotated frame.","rationale":"Read the full manuscript with attention to Theorem 1.2 and its proof. The central claim rests on the barrier constructions in Theorem 2.6 and Theorem 3.1, and the algebraic derivation of F[W] <= 0 appears internally consistent. The scaling argument in Theorem 3.2 is also sound. The only point that can break the proof is the coordinate rotation after choosing the nearest boundary point: the (a,eta)-type condition is not obviously invariant under changing to an arbitrary supporting normal. This matches the reader's weakest-assumption identification. However, the concern is not fatal as stated, because for a convex domain the metric projection of an interior point onto the boundary should land on a regular boundary point, where the supporting hyperplane is unique; then the rotated axis is forced to be the (a,eta)-type axis. The paper should state and prove this lemma. Separately, the final implication in Theorem 1.3 (\"which implies (1.6) for any delta\") is logically suspect because the exponent 2/(ab) is always below 1/a for b in (2,3); this does not affect Theorem 1.2 but should be corrected. Overall, the main theorem is plausible and the proof is likely repairable, so the conditional verdict stands.","tokens_in":11102,"tokens_out":45040,"duration_ms":459506,"concrete_test":"Prove or disprove the missing lemma: for a bounded convex Omega and any y in Omega, every nearest boundary point z has a unique supporting hyperplane. If the lemma is true, the rotation in Theorems 2.6 and 3.1 is justified and the gap closes. If false, exhibit a domain and a point y whose nearest boundary point z is a corner, then compute W on the boundary in the rotated coordinates to check whether (3.4) and F[W] <= 0 still hold; failure would refute the proof as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For each y in Omega, the proof of Theorem 3.1 picks a nearest boundary point z, rotates so the z-to-y direction is the x_n-axis, and asserts (3.2): Omega subset {x_n >= eta |x'|^a}. Definition 1.1 only guarantees such a containment in some coordinate system at z, not necessarily the one aligned with the metric-projection segment. At a corner with a wide normal cone, the containment for the particular direction z->y can fail (e.g., a wedge is (2,eta)-type only along its bisector, not along a face normal). The paper does not show that z can be chosen so that the relevant normal is the (a,eta)-type axis. This is a genuine gap in the written proof. It is likely repairable: for convex Omega and y in int(Omega), a nearest boundary point should have a unique supporting hyperplane (the tangent cone is a half-space), in which case the normal direction is unique and the (a,eta)-type containment aligns. But that lemma is absent.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the global Hölder regularity of the unique concave solution to the Dirichlet problem for the minimal graph equation in hyperbolic space over a bounded convex domain Ω⊂R^n. The authors introduce the notion of (a,η)-type convex domains (Definition 1.1) and prove (Theorem 1.2) that if Ω is (a,η)-type with a∈[2,+∞], then the solution belongs to C^{ā}(Ω) with ā=max{1/a,1/(n+1)} and a Hölder norm bounded by a constant depending only on a, η, dΩ, and n; explicit constants are given at the endpoints a=2 and a=+∞. The proof is based on constructing explicit supersolutions using the invariance of the equation under translations and rotations: the case a=2 is handled via an enclosing-ball ('exterior sphere') argument, the case a∈(2,∞) via a power-type barrier, and the case a∈(n+1,∞] via a one-dimensional barrier. A local boundary estimate for (a,η)-type points with a∈[1,2) is also stated (Theorem 1.3).","tokens_in":11336,"tokens_out":46511,"duration_ms":426544,"significance":"If correct, Theorem 1.2 provides a clean, quantitative regularity theory for minimal graphs in hyperbolic space over nonsmooth convex domains, recovering and refining the C^{1/(n+1)} result of Han-Shen-Wang and Lin's C^{1/2} for uniformly convex domains. The explicit barrier construction and explicit constants are valuable, and the paper ships detailed computations that are largely reproducible. The main weakness is that two key steps in the written proofs are not fully justified: the coordinate-alignment assertion in Theorems 2.6 and 3.1, and the final exponent in the proof of Theorem 1.3.","major_comments":[{"comment":"In the proofs of Theorem 2.6 (Section 2) and Theorem 3.1 (Section 3, Eq. (3.2)), the authors choose z∈∂Ω with |y−z|=d_y and then, after a rotation taking the segment z y to the x_n-axis, assert that the domain satisfies the containment Ω⊂B_R(R e_n) (Theorem 2.6) or Ω⊂{x_n≥η|x'|^a} (Theorem 3.1). This assertion does not follow directly from the exterior sphere condition or from Definition 1.1, since those conditions only guarantee the containment for some coordinate axis at z, not necessarily the axis aligned with the metric projection direction. The missing step is a lemma: for y∈Ω and a nearest point z∈∂Ω, the point z is a regular boundary point with a unique supporting hyperplane. Indeed, the open ball B(y,d_y) is contained in Ω and touches ∂Ω at z, so the tangent cone of Ω at z contains the tangent halfspace of that ball and hence must be a halfspace; consequently the (a,η)-type axis (or the exterior-sphere center direction) at z is forced to coincide with the direction y−z. This argument should be supplied before Eq. (3.2) and in the proof of Theorem 2.6. Without it, the barrier comparison is unjustified for interior points whose nearest boundary point is a corner.","section":"Theorem 2.6 and Theorem 3.1, around Eq. (3.2)"},{"comment":"The proof of Theorem 1.3 concludes u(0,x_n) ≤ x_n^{2/(ab)} for arbitrary b∈(2,3). Since b>2, the exponent 2/(ab) is strictly less than 1/a. For small x_n, x_n^{1/a+δ} is strictly smaller than x_n^{2/(ab)} for every δ>0, so the obtained bound is weaker than the claimed estimate (1.6) and cannot imply it. The direction of the implication is reversed: a bound with a larger exponent would imply one with a smaller exponent, not vice versa. Thus Theorem 1.3 is not proven as stated. The authors should either construct a barrier yielding an exponent at least 1/a+δ, or revise the statement of Theorem 1.3 to an exponent that the construction actually supports. This is a load-bearing issue for the local regularity theorem.","section":"Section 4, end of proof of Theorem 1.3"}],"minor_comments":[{"comment":"The word 'hyperbolic' is misspelled as 'hyp erbolic' in the abstract.","section":"Abstract"},{"comment":"'Chaplying gas' should be 'Chaplygin gas' in the introduction and in reference [9].","section":"Introduction and reference [9]"},{"comment":"There are extra unmatched parentheses in '(1/ε(a,η,dΩ,n)))^{1/a}'; the intended expression is '(1/ε)^{1/a} x_n^{1/a}'.","section":"Eq. (3.9)"},{"comment":"The statement 'U > 0 on ∂Ω' should be 'U ≥ 0 on ∂Ω', since U=0 on the part of the boundary with x_n=0.","section":"Theorem 3.2 proof"},{"comment":"The phrase 'Since (1,η)-type point is of course (1+ε, η(ε))-type point for any ε>0' should define η(ε); as written, η(ε) is not specified.","section":"Section 4, first paragraph"},{"comment":"The sign convention in '0 ≥ γ1(t) ≥ γ2(t)' is unclear; the geometric meaning of the heights should be stated more explicitly.","section":"Lemma 2.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The main theorem (Theorem 1.2) is likely correct, but the proof requires a nontrivial convex-geometry lemma to justify the coordinate alignment in Theorems 2.6 and 3.1; that lemma is readily supplied. More concerning is the error in Theorem 1.3: the barrier exponent is strictly below 1/a, so the claimed δ-improvement is unsupported and I suspect the statement is false. I recommend major revision rather than rejection because the central result is defensible, but the authors should be asked to either provide a correct proof of Theorem 1.3 or weaken the theorem accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that minimal graphs in hyperbolic space over (a,eta)-type convex domains are C^{bar a}, with bar a = max{1/a, 1/(n+1)}. The endpoints a=2 and a=+infinity were known, but the intermediate exponents are new, and the supersolution construction from rotational invariance is a genuinely different and clean approach. The explicit constants are nice, and the local estimate for a in [1,2) is a useful addition.\n\nThe core barrier computations are largely correct. I checked the algebra in Theorem 3.1: the cancellation at b=2 works, and the choice of small epsilon makes the supersolution valid. Theorem 3.2 for the general convex case also checks out. The paper is honest about what is optimal and does not oversell.\n\nThe reader's stress-test flagged the coordinate-alignment step in The proofs of Theorems 2.6 and 3.1. As written, the paper simply rotates so the segment from the nearest boundary point z to y is the x_n-axis and then asserts the containment Omega subset {x_n >= eta |x'|^a} or Omega subset B_R(R e_n). That assertion is not immediate. But on a careful read, the concern does not land. Since z is a nearest-boundary point of an interior point y, the open ball B(y,d_y) is contained in Omega and tangent at z. The tangent cone of that ball at z is a half-space, and because Omega is convex, its tangent cone at z contains that half-space and is also contained in some half-space, hence is exactly a half-space. So z is a regular boundary point with a unique supporting hyperplane, and that hyperplane must be the one perpendicular to z-y. Therefore any exterior sphere or (a,eta)-type containment, which must be attached to a supporting normal, is automatically aligned with the chosen axis. The paper should state this one-sentence lemma; its absence is a minor gap in exposition, not a flaw in the argument.\n\nMinor issues: a few algebraic typos (for example, the sign in one of the J_i terms before (3.8) and the notation in the constant for a=2), and the comparison-principle step could be spelled out. The citation pattern is fine; self-citation to [6] is for the (a,eta) framework and two supporting lemmas, which are independent of the main result.\n\nWho gets value: anyone working on boundary regularity for fully nonlinear or geometric elliptic equations, especially hyperbolic minimal graph problems. The method is transferable.\n\nThis deserves serious refereeing. I would accept after a minor revision that adds the regularity lemma for nearest boundary points and cleans up the typos.","headline":"The Holder regularity result is genuine and the barrier proof is basically sound; the coordinate-alignment concern is a minor omission, not a fatal gap.","tokens_in":11844,"tokens_out":20567,"would_cite":true,"duration_ms":195887,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J93","35B65","35J25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the global Hölder exponent of a minimal graph over a bounded convex domain in hyperbolic space is controlled by the domain's convexity type and not by its smoothness.","keywords":["minimal graphs","hyperbolic space","Dirichlet problem","global Hölder regularity","convex domain","(a,eta)-type domain","supersolution","boundary regularity"],"falsifier":"Solve the Dirichlet problem (1.1) on a lens-shaped convex domain formed by intersecting two disks of radius $R$, and test the bound $u(y) \\le 2\\sqrt{2 R d_\\Omega}\\sqrt{d_y}$ at points on the inward normal from one of the corners; exceeding the bound at any point would falsify the $a=2$ case of Theorem 1.2.","tokens_in":10883,"feed_emoji":"📐","tokens_out":15469,"duration_ms":124683,"temperature":0.7,"pith_summary":"This paper determines the exact global Hölder regularity of the solution to the Dirichlet problem for the equation of minimal graphs over bounded convex domains in hyperbolic space. The main theorem states that when the domain is of $(a,\\eta)$-type with $2 \\le a \\le +\\infty$, the unique concave solution belongs to $C^{\\bar a}(\\Omega)$, where $\\bar a = \\max\\{1/a, 1/(n+1)\\}$, with constants depending only on $a$, $\\eta$, the diameter, and the dimension. This shows that the regularity is controlled by the convexity of the boundary alone and is independent of its smoothness. The proof constructs explicit supersolutions from the translation- and rotation-invariance of the equation and uses the comparison principle, and it includes sharp endpoint constants for the strictly convex and flat-boundary cases.","feed_headline":"Convexity alone dictates minimal-graph Hölder exponents","feed_subtitle":"Sharp global regularity with explicit constants for hyperbolic minimal graphs over convex domains.","key_machinery":"The load-bearing device is a family of explicit supersolutions $W(x) = ((x_n/\\varepsilon)^{2/a} - |x'|^2)^{1/b}$, with $b=2$ in the main range, used in combination with the comparison principle. With a boundary point moved to the origin and the vertical axis aligned with the nearest-point segment, the $(a,\\eta)$-type containment $\\Omega \\subset \\{x_n \\ge \\eta |x'|^a\\}$ ensures $W \\ge u$ on the boundary, and a direct computation of the operator $F[W]$ shows $F[W] \\le 0$ for a sufficiently small $\\varepsilon$. The inequality $u \\le W$ then gives $u(y) \\le C d_y^{1/a}$ along the vertical axis, and Lemma 2.4 (a concave function vanishing on the boundary with $d^\\alpha$ decay is $C^\\alpha$) turns this into the global Hölder estimate. The endpoint cases use different barriers: for $a=2$, the explicit ball solution $U(x)=\\sqrt{R^2-|x|^2}$, available through the exterior sphere condition; for $a=+\\infty$, the one-variable supersolution $U(x)=(n+1)^2 x_n^{1/(n+1)} - x_n^{2-1/(n+1)}$ yields the universal $C^{1/(n+1)}$ estimate.","core_discovery":"Theorem 1.2 is the central claim: if $\\Omega \\subset \\mathbb{R}^n$ ($n \\ge 2$) is a bounded convex $(a,\\eta)$-type domain with $a \\in [2,+\\infty]$, then the concave solution $u$ of (1.1) satisfies $u \\in C^{\\bar a}(\\Omega)$ and $|u|_{C^{\\bar a}(\\Omega)} \\le C(a,\\eta,d_\\Omega,n)$, with $\\bar a = \\max\\{1/a, 1/(n+1)\\}$. The endpoint constants are explicit: $C = 2\\sqrt{2 R d_\\Omega}$ for $a=2$ ($R$ the exterior sphere radius) and $C = 2(n+1)^2 d_\\Omega^{1/(n+1)}$ for $a=+\\infty$. The theorem is optimal, since the exponent cannot be larger than $1/2$ for $(2,\\eta)$-type domains or larger than $1/(n+1)$ for $(+\\infty,\\eta)$-type domains whose boundary contains flat pieces. A separate local result, Theorem 1.3, gives the decay $|u(x)| \\le C |x|^{1/(a+\\delta)}$ along the normal direction near an $(a,\\eta)$-type boundary point for $a \\in [1,2)$.","pith_inferences":["The explicit supersolution construction is geometric rather than analytic, so the same barrier argument is likely to transfer to other translation- and rotation-invariant elliptic Dirichlet problems that admit a comparison principle.","Theorem 1.3 suggests a plausible global counterpart: a bounded convex domain whose boundary is $(a,\\eta)$-type only at finitely many corner points, with $a<2$, might still have a global exponent dictated by the sharpest corner; the paper does not prove this.","The $a=+\\infty$ constant $2(n+1)^2 d_\\Omega^{1/(n+1)}$ is likely not sharp, and the optimal constant could be found by improving the one-variable supersolution $U$; this is a testable optimization problem.","For numerical applications, the result means that replacing a smooth convex boundary by a polygonal one does not change the predicted worst-case Hölder exponent, which may guide the design of approximation schemes for minimal-surface-type equations."],"forward_implications":["For every $(2,\\eta)$-type convex domain, including nonsmooth ones, the minimal graph solution is globally $C^{1/2}$ with the explicit constant $2\\sqrt{2 R d_\\Omega}$.","For every bounded convex domain, the solution is globally $C^{1/(n+1)}$ with the explicit constant $2(n+1)^2 d_\\Omega^{1/(n+1)}$, giving a quantitative universal regularity estimate.","For $(a,\\eta)$-type domains with $2 < a \\le n+1$, the global exponent is $1/a$; for $a > n+1$ it remains $1/(n+1)$, so boundary convexity controls regularity only down to the universal floor.","The optimality statements mean that no convexity-based improvement can push the exponent above $1/2$ for strictly convex $(2,\\eta)$-type domains or above $1/(n+1)$ for domains with flat boundary pieces.","Because the constants depend only on the $(a,\\eta)$-type geometry, the same Hölder estimates hold for polyhedral approximations of smooth convex domains."],"supporting_citations":[{"why":"Establishes the prior optimal regularity theorem for bounded convex domains, including the $C^{1/(n+1)}$ result and the sharpness remark that the present theorem refines and quantifies.","marker":"[4]"},{"why":"Introduces the $(a,\\eta)$-type classification of boundary convexity and the key lemma converting boundary decay into Hölder regularity; also supplies the equivalence between $(2,\\eta)$-type domains and the exterior sphere condition.","marker":"[6]"},{"why":"Proves existence, uniqueness, and interior regularity for the Dirichlet problem, and the $C^{1/2}$ boundary estimate for smooth strictly convex domains that the $a=2$ case generalizes to nonsmooth $(2,\\eta)$-type domains.","marker":"[7]"}],"fun_headline_variants":["Convexity alone settles optimal Holder exponents for minimal graphs","Optimal Holder exponents for minimal graphs: convexity suffices","Sharp global regularity of minimal graphs over convex domains","Explicit constants for optimal regularity in hyperbolic minimal graphs","Convexity alone fixes sharp Holder exponents for minimal graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that for each boundary point and a nearest interior point, the $(a,\\eta)$-type containment can be written in coordinates whose vertical axis runs along the segment between those two points, even though the definition of $(a,\\eta)$-type only guarantees that such a containment holds in some coordinate system.","fun_headline_variants_meta":{"raw":{"variants":["Convexity alone settles optimal Holder exponents for minimal graphs","Optimal Holder exponents for minimal graphs: convexity suffices","Sharp global regularity of minimal graphs over convex domains","Explicit constants for optimal regularity in hyperbolic minimal graphs","Convexity alone fixes sharp Holder exponents for minimal graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001351,"raw_usage":{"total_tokens":5455,"prompt_tokens":884,"completion_tokens":4571,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":4492}},"tokens_in":500,"tokens_out":4571,"duration_ms":27908,"temperature":1.0,"reasoning_tokens":4492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:49:19.754041+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the Dirichlet problem (1.1) on a lens-shaped convex domain formed by intersecting two disks of radius $R$, and test the bound $u(y) \\le 2\\sqrt{2 R d_\\Omega}\\sqrt{d_y}$ at points on the inward normal from one of the corners; exceeding the bound at any point would falsify the $a=2$ case of Theorem 1.2.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the prior optimal regularity theorem for bounded convex domains, including the $C^{1/(n+1)}$ result and the sharpness remark that the present theorem refines and quantifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the $(a,\\eta)$-type classification of boundary convexity and the key lemma converting boundary decay into Hölder regularity; also supplies the equivalence between $(2,\\eta)$-type domains and the exterior sphere condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves existence, uniqueness, and interior regularity for the Dirichlet problem, and the $C^{1/2}$ boundary estimate for smooth strictly convex domains that the $a=2$ case generalizes to nonsmooth $(2,\\eta)$-type domains."}],"review_version":1}