Invariant hypersurfaces with linear prescribed mean curvature

Antonio Bueno; Irene Ortiz

REVIEW 3 major objections 4 minor 19 references

Invariant hypersurfaces with linear prescribed mean curvature

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read All rotational Hλ-hypersurfaces in Euclidean space are classified by one parameter, λ.

desk verdict Useful classification paper with a real but localized bug in the explicit formulas; the rotational part is worth refereeing. read the letter →

arxiv 1908.07378 v1 pith:HXGWGEUO submitted 2019-08-20 math.DG

Antonio Bueno , Irene Ortiz This is my paper · ORCID

classification math.DG MSC 53A1053C4234C0534C40

keywords prescribedmeancurvaturerotationalhypersurfaceweightedphaseplaneanalysisself-translatingsolitonautonomoussystemconstantcylinderGaussmap

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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 $n\ge 2$.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

The 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.

Second 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.

The 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.

Who 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.

Referee Report

3 major / 4 minor

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.

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 (3)

[Section 2, Theorem 2.1, cases λ>v_{n+1} and λ=v_{n+1}] 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 2, Theorem 2.1, case λ<v_{n+1}, θ(0)=π] 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.
[Lemma 4.1 and Proposition 4.2(3)] 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.

minor comments (4)

[Section 4, page 13, paragraph before Figure 8] 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.
[Throughout Section 4] 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 2, displayed θ-formulas] The word 'arccotg' should be 'arccot' or 'arccotangent' for consistency with the other formulas.
[Abstract and introduction] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rotational and cylindrical classifications are derived from the ODE and prior independent results; noted gaps are missing proofs/formula defects, not self-reference.

full rationale

The claimed classification is not circular. Hλ-hypersurfaces are defined by equation (1.2), and the cylindrical and rotational reductions are obtained by substituting the geometric ansatz into the standard mean-curvature formula: equations (2.2) and (3.1)-(3.3) are derived from the geometry, not assumed. The subsequent classifications are phase-plane consequences: the equilibrium and eigenvalues are computed from the linearization (4.1), the monotonicity regions come from the nullcline Γε in (3.4), and the orbit-endpoint assertions rest on Lemma 3.2, a general existence/uniqueness result taken from [BGM2] and ultimately supported by the Dirichlet problem in [Mar], rather than on the Hλ classification being proved. Lemma 4.1 cites an external n=2 theorem [Lop] and states that the proof extends; while the n>2 argument is not written out, and Theorem 2.1 contains an apparent normalization defect in the printed formulas, these are correctness and completeness concerns, not cases where a conclusion is manufactured from its own input. No fitted parameter is relabeled as a prediction, and no defined quantity collapses into the target result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new entities. Its central claims rest on classical cylinder theorems, on the phase-plane framework imported from [BGM2] and [Mar], on the rotational-axis alignment result from [Lop], and on the n-dimensional extension of López's no-closed-example lemma, which is asserted rather than proved.

assumptions (6)

standard math Complete hypersurfaces of constant curvature in R^{n+1} are flat, hence cylindrical over a plane curve (Liebmann, Hilbert, Hartman-Nirenberg).
Used in Section 2 to reduce constant-curvature Hλ-hypersurfaces to base curves α.
domain assumption For a rotational Hλ-hypersurface, the density vector v must be parallel to the rotation axis ([Lop, Proposition 4.3]).
Imported at the start of Section 3; allows the normalization v = e_{n+1}.
domain assumption Existence and uniqueness of phase-plane orbits with endpoints (0, ±1) on the boundary (Lemma 3.2, from [BGM2, Corollary 2.4] and [Mar, Corollary 1]).
Used to start every axis-intersecting rotational hypersurface in Section 4.
domain assumption There are no closed Hλ-hypersurfaces (Lemma 4.1); for n = 2 this is [Lop], and for n > 2 the paper asserts the proof extends without giving it.
Rules out the glued compact orbit in Proposition 4.2, item 3; not proved in this paper for general n.
standard math Standard stable-manifold and linearization theory for planar autonomous systems.
Used in Section 4 to describe orbits near e0 depending on λ versus √((n−1)/2).
standard math Mean curvature comparison principle for graphs.
Used in Proposition 4.2 to contradict the existence of an entire convex graph with bounded mean curvature.

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Pith. "Pith review of Invariant hypersurfaces with linear prescribed mean curvature." pith.science (2026). https://pith.science/paper/HXGWGEUO

@misc{pith2026190807378,
  author       = {Pith},
  title        = {Pith review of: Invariant hypersurfaces with linear prescribed mean curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXGWGEUO}},
  note         = {Machine review of arXiv:1908.07378}
}

abstract

Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

Figures

Figures reproduced from arXiv: 1908.07378 by the authors.

**Figure 1.** The profile curve for the case λ > vn+1. Here, n = 2, vn+1 = 1 and λ = 2. 2. Case λ = vn+1. 2.1. Either α(s) is a horizontal straight line parametrized by x(s) = −s, z(s) = c0, c0 ∈ R, θ(s) = π, or 2.2. its explicit coordinates are x(s) = −s + n 2 arctan(ns), z(s) = 1 n log(1 + n 2 s 2 ). The image of the angle function θ(s) in the circle S 1 is S 1−{(0, −1)}. The z(s)-coordinate decreases until reaching a minimum a… view at source ↗

**Figure 2.** The profile curves for the case λ = vn+1. Here, n = 2, vn+1 = 1 and λ = 1. 3. Case λ < vn+1. 3.1. Either α(s) is a straight line parametrized by x(s) = (cos θ0)s, z(s) = ±(sin θ0)s, θ(s) = θ0, where θ0 is such that λ + vn+1 cos θ0 = 0, or 3.2. if θ(0) = 0, its explicit coordinates are x(s) = −λs + 2 nvn+1 arctan qvn+1+λ vn+1−λ tanh n 2 q v 2 n+1 − λ2s , z(s) = 1 nvn+1 log −λ + cosh n q v 2 n+1 − λ2s . I… view at source ↗

**Figure 3.** Left: the profile curves for the case λ < vn+1. In blue, the case θ(0) = 0; in orange, the case θ(0) = π. Here, n = 2, vn+1 = 1 and λ = 1/2. Right: the values of θ(s) in S 1 of each curve. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: The linearized of system (3.3) depending on the values of λ > 0 and the behavior of its orbits. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: The phase planes Θε, ε = ±1 for λ > 1, their monotonicity regions and two orbits following the motion at each monotonicity region. Our first goal is to describe the rotational Hλ-hypersurfaces that intersect orthogonally the axis of rotation. By Lemma 3.2 there is an o…

**Figure 6.** Figure 6: Left: the phase plane Θ1 and the orbits γ+ and γ−. Right: the corresponding arc-length parametrized curves α+ and α−. Let us study in more detail the behavior of both orbits γ+ and γ− in Θ1. Proposition 4.2 Let us consider the orbits γ+ and γ− in the phase plane Θ1 as …

**Figure 7.** Figure 7: Left: the configurations that cannot happen in Θ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: left and right, the continuous plot. • if λ < √ n − 1/2, then γ+ converges to e0 directly, that is without spiraling around it. As a consequence, α 0 + is never vertical and thus Σ+ is a strictly convex graph that converges to C [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: Left: the phase planes Θ1 and Θ−1 and the orbit γ−. Right: the corresponding arc-length parametrized curve α−. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: , right). Θ1 Θ−1 γ b (r0, 0) b (r1, 1) (r2, −1) b b (r2, −1) b (r3, 1) b (r3, 1) [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: The phase planes Θε, ε = ±1 for λ = 1, their monotonicity regions and two orbits following the motion at each monotonicity region. We first study the rotational H1-hypersurfaces intersecting the axis of rotation. For this purpose, we must begin by pointing out that a …

**Figure 12.** Figure 12: Top left: the configuration that cannot happen in Θ [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: Left: the phase planes Θ1 and Θ−1 and the orbits γ+ and γ−. Right: the corresponding arc-length parametrized curves α+ and α−. Finally, we analyze the Hλ-hypersurfaces non-intersecting the axis of rotation. For that, let γ be an orbit in Θ1 passing through a point (x,…

**Figure 14.** Figure 14: Left: The phase planes Θ1 and Θ−1 and the two possible configurations for the orbit γ. Center and right: the two corresponding arc-length parametrized curves α. To finish, we summarize the discussion carried on along this section in two classification results of the r…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad Univ

A.D. Alexandrov, Uniqueness theorems for surfaces in the large, I, Vestnik Leningrad Univ. 11 (1956), 5--17. (English translation: Amer. Math. Soc. Transl. 21 (1962), 341--354)

work page 1956

[2] [2]

Bayle, A

V. Bayle, A. Cañete, F. Morgan, C. Rosales, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Diff. Equations 31 (2008), 27--46

work page 2008

[3] [3]

Bueno, The Björling problem for prescribed mean curvature surfaces in 3 , Ann

A. Bueno, The Björling problem for prescribed mean curvature surfaces in 3 , Ann. Glob. Ann. Geom. 56 2019, 87--96

work page 2019

[4] [4]

Bueno, Half-space theorems for properly immersed surfaces in 3 with prescribed mean curvature, Ann

A. Bueno, Half-space theorems for properly immersed surfaces in 3 with prescribed mean curvature, Ann. Mat. Pur. Appl. DOI: 10.1007/s10231-019-00886-1

work page doi:10.1007/s10231-019-00886-1

[5] [5]

The global geometry of surfaces with prescribed mean curvature in $\mathbb{R}^3$

A. Bueno, J.A. Gálvez, P. Mira, The global geometry of surfaces with prescribed mean curvature in ^3 , preprint. arxiv:1802.08146

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