{"id":"1de4fd8f-e930-410b-a597-8f3d01476814","arxiv_id":"1908.07378","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For hypersurfaces with mean curvature H = ⟨η, v⟩ + λ, the paper gives explicit parametrizations of cylindrical flat examples and a complete classification of rotational examples.","lead":"This paper classifies hypersurfaces in Euclidean space whose mean curvature equals a fixed constant plus a linear function of the surface normal. The classification covers flat cylindrical examples with explicit formulas and rotational examples, connecting to weighted mean curvature and translating solitons.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1's explicit parametrizations are false unless v_{n+1}=1; at s=0 the printed x(s) has derivative v_{n+1} instead of 1, so the advertised integration result needs correction.","rationale":"The reader's chosen weak point is Proposition 4.2's reliance on Lemma 4.1. I checked this first because it is the hinge of the uniqueness claim. The lemma is true and has a short proof using the standard identities ∫_Σ Hη dA=0 and ∫_Σ η dA=0 for any closed oriented hypersurface: dotting the first with v and substituting H=⟨η,v⟩+λ gives ∫⟨η,v⟩²=0, hence v∈TΣ at every point; the constant vector field v then generates a global flow on the compact hypersurface whose orbits are affine lines p+tv, which are unbounded, a contradiction. So the 'extends easily' assertion is not a correctness risk, though the paper should have included the proof. The actual concrete error I found is in Theorem 2.1, an advertised main result. Because v is a unit vector with only v_{n+1} normalized to be nonzero, v_{n+1} is generically not 1; the printed x,z formulas are not solutions of (2.2) in that generic case, as the s=0 derivative shows. This is not a matter of convention, since system (2.2) itself retains v_{n+1}. Section 4 does not use Theorem 2.1, so I have no substantive objection to the rotational classification from my reading; the phase-plane claims are qualitative but coherent, and the existing CONDITIONAL verdict remains appropriate provided the parametrization theorem is corrected and the proof of Lemma 4.1 is supplied.","tokens_in":17325,"tokens_out":25470,"duration_ms":240358,"concrete_test":"Take v_{n+1}=c∈(0,1), e.g. c=1/2, λ=2, n=2, and evaluate the printed x(s) from Theorem 2.1 at s=0: since θ(0)=0, system (2.2) requires x'(0)=1, but the formula differentiates to c=1/2. Recompute the integrals from x'=cosθ and θ'=n(c cosθ+λ): x(s)=(θ(s)/n-λs)/c up to the chosen origin, and z(s)=-(1/(n c))log(c cosθ(s)+λ)+const. If the authors' formulas are amended by these factors, and with the correct 2/n coefficient in the λ=c case, the check passes; otherwise Theorem 2.1 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The rotational classification (Theorems 4.3 and 4.4) appears internally coherent, but I cannot sign off on the paper as a whole because Theorem 2.1 is false as stated. In Section 2, after a rotation the density vector is v=(0,v_2,...,v_{n+1}) with v_{n+1}\\ne 0, and no normalization v_{n+1}=1 is made (v is unit, so v_{n+1}\\in(-1,1)). The solutions of (2.2) are integrated with θ(0)=0. The printed formula in case λ>v_{n+1}, x(s)=-λs+(2/n)\\arctan(\\sqrt{(λ+v_{n+1})/(λ-v_{n+1})}\\tan(n/2\\sqrt{λ^2-v_{n+1}^2}s)), has x'(0)=v_{n+1}, while (2.2) forces x'(0)=cosθ(0)=1. Direct integration gives x(s)=(θ(s)/n-λs)/v_{n+1}; the z-formula has the same defect. The λ=v_{n+1} case also prints n/2 instead of 2/n even for v_{n+1}=1. Lemma 4.1 is not, in my view, a load-bearing gap: for a closed H_λ-hypersurface, ∫H⟨η,v⟩=0 and ∫⟨η,v⟩=0 give ⟨η,v⟩=0 pointwise, so v is tangent and the affine lines p+tv stay in a compact hypersurface, impossible. The missing n-dimensional proof is routine, though it should be supplied. Thus the concrete defect is the incorrect advertised parametrization in Theorem 2.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies oriented hypersurfaces Σ^n in R^{n+1} whose mean curvature satisfies H_Σ(p)=⟨η_p,v⟩+λ, the class of H_λ-hypersurfaces, and connects them to weighted mean curvature, the volume-preserving mean curvature flow, and translating solitons. Section 2 treats complete flat (cylindrical) H_λ-hypersurfaces and claims an explicit classification of their base curves by solving system (2.2). Section 3 sets up the phase-plane analysis for rotational H_λ-hypersurfaces, following [BGM2], and Section 4 gives the main classification: Theorem 4.3 for rotational hypersurfaces intersecting the rotation axis and Theorem 4.4 for those not intersecting it. The claimed classification is that Σ_+ is always properly embedded, simply connected, and asymptotic to the cylinder C((n−1)/(λn)), while Σ_− is a horizontal hyperplane for λ=1, an entire strictly convex graph for λ<1, and a properly immersed surface with infinitely many self-intersections for λ>1; every non-axis-intersecting example is properly immersed, diffeomorphic to S^{n−1}×R, with one end asymptotic to the cylinder and the other end either self-intersecting at infinity or a graph outside a compact set.","tokens_in":17623,"tokens_out":18748,"duration_ms":180766,"significance":"If the results are correct, the rotational classification is a substantial contribution: it extends López's n=2 classification to all dimensions, uses a coherent phase-plane framework, and gives falsifiable geometric conclusions about embeddedness, self-intersections, and asymptotics. The paper also correctly identifies the relation between H_λ-hypersurfaces, weighted mean curvature, and the geometric flow (1.4). The overall architecture of Section 4 is coherent, and the rotational classification does not appear to depend on the explicit formulas of Theorem 2.1. However, Theorem 2.1, which is one of the two advertised main results, is false as stated for general v_{n+1}: the printed parametrizations do not solve system (2.2) unless normalization conditions that are not made are imposed. The paper therefore cannot be accepted in its current form, but the defects are localized and fixable.","major_comments":[{"comment":"The printed explicit coordinates do not satisfy system (2.2) for general v_{n+1}. In the case λ>v_{n+1}, differentiating the displayed x(s) gives x'(0)=v_{n+1}, whereas (2.2) together with θ(0)=0 forces x'(0)=cos 0=1. The correct integration is x(s)=(θ(s)/n−λs)/v_{n+1}; for the λ=v_{n+1} case the factor n/2 in front of arctan(ns) should be 2/(n v_{n+1}) with argument n v_{n+1}s, and the z(s) formula likewise carries a missing factor 1/v_{n+1}. These are not cosmetic issues: Theorem 2.1 is the paper's advertised classification of cylindrical flat H_λ-hypersurfaces, and as printed it is false for v_{n+1}≠1 (and, in the λ=v case, even for v_{n+1}=1 unless n=2).","section":"Section 2, Theorem 2.1, cases λ>v_{n+1} and λ=v_{n+1}"},{"comment":"The displayed x(s) in this case also fails the initial condition: direct differentiation gives x'(0)=−λ−(v_{n+1}+λ)/v_{n+1}, which is not the required cos π=−1. The root inside the arctan should be √((v_{n+1}−λ)/(v_{n+1}+λ)), matching the θ(s)=2 arccot(...) formula displayed before the theorem, not the reciprocal √((v_{n+1}+λ)/(v_{n+1}−λ)) that is printed. Thus this case of Theorem 2.1 is incorrect as stated.","section":"Section 2, Theorem 2.1, case λ<v_{n+1}, θ(0)=π"},{"comment":"Lemma 4.1 asserts that no closed H_λ-hypersurface exists in any dimension, but the text only cites the n=2 proof in [Lop] and says the proof 'can be easily extended'. This lemma is load-bearing: Proposition 4.2(3) rules out the configuration x_+=x_− by gluing γ_+ and γ_− into a closed rotational sphere and invoking Lemma 4.1. The authors should supply the n-dimensional proof in the paper or give a reference that covers all dimensions. A short proof is available: from the first variation formula and the divergence theorem one gets ∫_Σ H_Σ⟨η,v⟩=0 and ∫_Σ⟨η,v⟩=0; since H_Σ=⟨η,v⟩+λ, this yields ∫_Σ⟨η,v⟩^2=0, so v is tangent to Σ, and the complete lines p+tv would then lie in the compact hypersurface, a contradiction.","section":"Lemma 4.1 and Proposition 4.2(3)"}],"minor_comments":[{"comment":"The text says 'Σ_+ is a strictly convex graph that converges to C(n−1/n)'; in the λ>1 subsection this should be C((n−1)/(λn)), and the notation C(r) should be defined at its first use.","section":"Section 4, page 13, paragraph before Figure 8"},{"comment":"The expressions 'λ>√n−1/2' and 'λ<√n−1/2' should be typeset as λ>√((n−1)/2) and λ<√((n−1)/2), since the current notation is ambiguous.","section":"Throughout Section 4"},{"comment":"The word 'arccotg' should be 'arccot' or 'arccotangent' for consistency with the other formulas.","section":"Section 2, displayed θ-formulas"},{"comment":"The abstract says the paper obtains 'explicit parametrizations of constant curvature hypersurfaces'; the body treats flat cylindrical H_λ-hypersurfaces. The wording should be aligned to avoid suggesting a classification of all constant-mean-curvature hypersurfaces.","section":"Abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is Theorem 2.1, which is clearly incorrect as stated for general v_{n+1}; the rotational classification in Section 4 appears internally coherent and independent of those formulas. The authors should also clarify the status of the preprints [BGM1] and [BGM2], since several load-bearing statements are quoted from them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a legitimate paper, and the rotational classification (Theorems 4.3 and 4.4) is probably correct, but the explicit parametrizations in Theorem 2.1 are wrong as printed. Fix that and the missing proof of Lemma 4.1, and it is publishable.\n\nWhat is new: the paper completes the H_λ = ⟨η,v⟩+λ invariant hypersurface picture for cylindrical and rotational cases in arbitrary dimension. Theorem 2.1 aims to give explicit base curves; Theorems 4.3 and 4.4 classify rotational examples by whether they meet the axis. The n=2 case was known in López, and the phase-plane method comes from BGM2, but the n-dimensional rotational classification and the explicit parametrizations appear genuinely new. The connection to weighted mean curvature and to self-translating solitons is real, and the paper is honest about what it takes from BGM2, López, and Marquardt.\n\nThe main soft spot is Section 2. After rotating so v = (0,...,v_{n+1}) with v_{n+1} ≠ 0, no normalization v_{n+1}=1 is made. The printed θ(s) is fine, but the integrated x(s) in case λ > v_{n+1} is not. At s=0, x'(0) from the printed formula is v_{n+1}, while equation (2.2) requires x'(0)=cos θ(0)=1. Direct integration gives x(s) = (θ(s)/n - λs)/v_{n+1}, and the printed z(s) has the same defect. In the case λ = v_{n+1} the coefficient is also wrong: even for v_{n+1}=1 it should be (2/n) arctan(ns), not (n/2). So the advertised formulas need correction before anyone can use them.\n\nSecond soft spot: Lemma 4.1, the no-closed-H_λ-hypersurfaces statement, is asserted for all n with “the proof extends easily” from the n=2 case. The text uses it to rule out gluing γ+ and γ− into a closed rotational sphere in Proposition 4.2(3), so it is load-bearing. But the gap is small: for a closed H_λ-hypersurface, ∫ H⟨η,v⟩=0 and ∫⟨η,v⟩=0 are standard, giving ∫⟨η,v⟩²=0, hence v is tangent, and the affine lines p+tv stay in a compact hypersurface—impossible. The authors should write this out, but it is not a fatal flaw.\n\nThe phase-plane discussion is credible. I did not see an internal contradiction in Theorems 4.3 and 4.4; the eigenvalue threshold analysis is standard, and the comparison arguments in the λ=1 case are plausible. The citation pattern is fine: self-citations point to BGM1/BGM2, where the machinery was developed.\n\nWho should read this: people working on prescribed mean curvature, weighted CMC hypersurfaces, and translating solitons. It deserves a serious referee. I would send it out, not desk reject it, with a request to correct Theorem 2.1 and supply a full proof of Lemma 4.1. After that, I would expect it to be accepted.","headline":"Useful classification paper with a real but localized bug in the explicit formulas; the rotational part is worth refereeing.","tokens_in":18237,"tokens_out":4547,"would_cite":false,"duration_ms":44758,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53A10","53C42","34C05","34C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"All rotational Hλ-hypersurfaces in Euclidean space are classified by one parameter, λ.","keywords":["prescribed mean curvature","rotational hypersurface","weighted mean curvature","phase plane analysis","self-translating soliton","autonomous system","constant mean curvature cylinder","Gauss map"],"falsifier":"The decisive check is numerical: for $n=3$ and $\\lambda=2$, integrate system (3.3) from the two boundary conditions $(0,1)$ and $(0,-1)$ and compare the $x$-coordinates where each orbit first crosses the axis $y=0$; if they coincide, the two profiles glue into a closed rotational $H_\\lambda$-hypersurface, which Lemma 4.1 forbids. Any explicit closed $H_\\lambda$-hypersurface in $\\mathbb{R}^4$ would falsify the central classification.","tokens_in":17035,"feed_emoji":"📐","tokens_out":12233,"duration_ms":108238,"temperature":0.7,"pith_summary":"This paper aims to classify all complete hypersurfaces in Euclidean space whose mean curvature is prescribed as a linear function of the Gauss map, $H_\\lambda(\\eta)=\\langle\\eta,v\\rangle+\\lambda$. Such $H_\\lambda$-hypersurfaces have constant weighted mean curvature $\\lambda$ for the density $e^{\\langle x,v\\rangle}$, include self-translating solitons of mean curvature flow as the limit $\\lambda\\to 0$, and are critical points of a weighted area-minus-volume functional. The authors give explicit parametrizations of the flat cylindrical examples and a full classification, by the single parameter $\\lambda>0$, of the rotationally symmetric examples. A reader might care because the classification reduces a global geometric question to the phase-plane analysis of a two-dimensional autonomous ODE system and extends the two-dimensional classification to every dimension $n\\ge 2$.","feed_headline":"A single parameter λ fully classifies rotational Hλ-hypersurfaces","feed_subtitle":"Depending on λ, each one is a hyperplane, a convex graph, or a surface spiraling into a cylinder.","key_machinery":"The central mechanism is the phase plane $\\Theta_\\varepsilon=(0,\\infty)\\times(-1,1)$ for the autonomous system $x'=y$, $y'=(n-1)(1-y^2)/x-n\\varepsilon(y+\\lambda)\\sqrt{1-y^2}$, where $\\varepsilon=\\pm 1$ records whether the profile's height is increasing or decreasing. An orbit $\\gamma(s)=(x(s),y(s))$ determines the profile curve $\\alpha(s)=(x(s),z(s))$ of a rotational $H_\\lambda$-hypersurface by integration; the unique equilibrium $e_0=((n-1)/(\\lambda n),0)$ corresponds to the flat cylinder $C((n-1)/(\\lambda n))$, and the curve $\\Gamma_\\varepsilon$ where $y'=0$ splits the plane into monotonicity regions. Lemma 3.2 gives the unique orbits ending at the boundary points $(0,\\pm 1)$, which represent profiles meeting the axis orthogonally, while Proposition 4.2 uses Lemma 4.1 (no closed $H_\\lambda$-hypersurfaces) to rule out the two axis-meeting orbits gluing into a sphere. The same ODE setup, integrated explicitly for the base curve of a cylindrical example, yields Theorem 2.1's parametrizations.","core_discovery":"The paper's central discovery is that, in $\\mathbb{R}^{n+1}$, an immersed oriented hypersurface whose mean curvature satisfies $H_\\Sigma(p)=\\langle\\eta_p,v\\rangle+\\lambda$ can be fully described from the single number $\\lambda$ once rotational symmetry is imposed. There are two geometric classes: profiles that meet the rotation axis orthogonally and profiles that do not. The axis-intersecting surface with upwards orientation is properly embedded, simply connected, and converges to the flat constant-mean-curvature cylinder $C((n-1)/(\\lambda n))$ of radius $(n-1)/(\\lambda n)$, intersecting it infinitely often when $\\lambda>\\sqrt{n-1}/2$, finitely often when $\\lambda=\\sqrt{n-1}/2$, and not at all when $\\lambda<\\sqrt{n-1}/2$. The downwards-oriented axis-intersecting surface is a horizontal hyperplane when $\\lambda=1$, a strictly convex entire graph when $\\lambda<1$, and a properly immersed surface with infinitely many self-intersections and unbounded distance to the axis when $\\lambda>1$. Every non-axis-intersecting rotational example is properly immersed and diffeomorphic to $S^{n-1}\\times\\mathbb{R}$, with one end asymptotic to the same cylinder and the other end either self-intersecting at infinity ($\\lambda>1$) or a graph outside a compact set ($\\lambda\\le 1$). The paper also classifies complete constant-curvature $H_\\lambda$-hypersurfaces, which must be flat and cylindrical, by explicit parametrizations of their base curves.","pith_inferences":["The proof of uniqueness in Theorem 4.3 leans on Lemma 4.1, whose higher-dimensional case is only asserted; if a closed $H_\\lambda$-hypersurface existed for $n\\ge3$, the two axis-meeting orbits could glue into a sphere and the classification would need revision. This is an inference about the proof's reliance, not a claim made in the paper.","The same end dichotomy—one end asymptotic to the cylinder, the other end a graph or an unbounded self-intersecting end—may be a property of the equation rather than of rotational symmetry, so non-rotational complete $H_\\lambda$-hypersurfaces are worth testing for the same behavior.","The explicit cylindrical parametrizations could seed numerical searches for periodic orbits of the rotational system in higher dimensions; a closed solution would appear as a periodic orbit gluing across the boundary, which Lemma 4.1 says cannot exist.","For $\\lambda=1$, the horizontal hyperplane solution invites a Bernstein-type question: whether every complete embedded $H_1$-hypersurface that is a graph over a horizontal hyperplane must itself be a hyperplane."],"forward_implications":["In every dimension $n\\ge2$, the rotational $H_\\lambda$ landscape is a one-parameter family: the value of $\\lambda$ alone decides whether an end is a properly embedded graph, a spiraling approach to a cylinder, or an unbounded self-intersecting immersion.","The axis-nonintersecting examples force the topology $S^{n-1}\\times\\mathbb{R}$ onto every complete rotational $H_\\lambda$-hypersurface that avoids the axis.","Because $H_\\lambda$-hypersurfaces are exactly the constant-weighted-mean-curvature surfaces for the density $e^{\\langle x,v\\rangle}$, the classified examples are explicit critical points of the weighted area-minus-volume functional.","For $\\lambda<1$, the downwards-oriented axis-intersecting surface is a strictly convex entire graph, so the classification yields entire graphical solutions of the prescribed-mean-curvature equation.","For $\\lambda=1$, the only downwards-oriented axis-intersecting rotational example is a horizontal hyperplane, isolating that parameter value as a rigidity point."],"supporting_citations":[{"why":"Supplies the phase-plane system, monotonicity regions, and Corollary 2.4 used to construct the unique orbits ending at the axis.","marker":"[BGM2]"},{"why":"Provides the n=2 proof that no closed Hλ-hypersurface exists, which Lemma 4.1 extends, and the n=2 classification that Theorems 4.3–4.4 recover.","marker":"[Lop]"},{"why":"Guarantees the existence of rotational Hλ-hypersurfaces meeting the axis orthogonally via the Dirichlet problem, used in Lemma 3.2.","marker":"[Mar]"},{"why":"Introduces the prescribed-mean-curvature equation (1.1) and the Hλ class (1.2) that the paper studies.","marker":"[BGM1]"},{"why":"Establishes that constant weighted mean curvature is equivalent to being a critical point of weighted area minus λ volume.","marker":"[BCMR]"},{"why":"Frames the λ=0 case as self-translating solitons of mean curvature flow, connecting the class to known theory.","marker":"[Ilm]"}],"fun_headline_variants":["λ is the only key to rotational Hλ-hypersurfaces","Every rotational Hλ-hypersurface boils down to λ","Full λ-classification of rotational Hλ-hypersurfaces","One λ controls hyperplanes, graphs, and spiral Hλ","λ alone sorts all rotational Hλ: hyperplane, graph, or spiral"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification depends on Lemma 4.1, which says no closed $H_\\lambda$-hypersurface exists in any dimension; the paper gives the proof only for $n=2$ and asserts that it extends easily to higher dimensions, so a closed example for $n\\ge3$ would collapse the uniqueness part of the classification.","fun_headline_variants_meta":{"raw":{"variants":["λ is the only key to rotational Hλ-hypersurfaces","Every rotational Hλ-hypersurface boils down to λ","Full λ-classification of rotational Hλ-hypersurfaces","One λ controls hyperplanes, graphs, and spiral Hλ","λ alone sorts all rotational Hλ: hyperplane, graph, or spiral"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000796,"raw_usage":{"total_tokens":3507,"prompt_tokens":953,"completion_tokens":2554,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":2462}},"tokens_in":569,"tokens_out":2554,"duration_ms":17494,"temperature":1.0,"reasoning_tokens":2462,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:20:19.381725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is numerical: for $n=3$ and $\\lambda=2$, integrate system (3.3) from the two boundary conditions $(0,1)$ and $(0,-1)$ and compare the $x$-coordinates where each orbit first crosses the axis $y=0$; if they coincide, the two profiles glue into a closed rotational $H_\\lambda$-hypersurface, which Lemma 4.1 forbids. Any explicit closed $H_\\lambda$-hypersurface in $\\mathbb{R}^4$ would falsify the central classification.","supporting_citations":[],"review_version":1}