arxiv: 2605.11649 · v1 · submitted 2026-05-12 · 🧮 math.DG

Recognition: 2 theorem links

· Lean Theorem

Homothetical surfaces with constant mean curvature in hyperbolic space

Rafael Belli, Rafael L\'opez

Pith reviewed 2026-05-13 01:11 UTC · model grok-4.3

classification 🧮 math.DG

keywords homothetical surfacesconstant mean curvaturehyperbolic spaceparabolic surfacesupper half-space modelsurface classificationminimal surfacesCMC surfaces

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The pith

Any homothetical surface with constant mean curvature in hyperbolic 3-space must be parabolic, forcing either phi or psi constant in the product z equals phi of x times psi of y.

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

The paper classifies homothetical surfaces of constant mean curvature H in hyperbolic 3-space. These surfaces are graphs of the product form z equals phi of x times psi of y in the upper half-space model. The central result is that constant H forces the surface to be parabolic, so one of the two functions must be constant. This holds for minimal surfaces where H equals zero, for H squared not equal to one, and for the critical case H squared equals one. The classification extends earlier work on parabolic constant-mean-curvature surfaces by showing no non-parabolic examples exist inside this product family.

Core claim

We classify all homothetical surfaces with constant mean curvature H in the hyperbolic space H^3. Using the upper half-space model with standard coordinates (x,y,z), these surfaces are defined by the relation z = phi(x)psi(y), where phi and psi are smooth functions of one variable. We demonstrate that any such surface is necessarily parabolic, meaning that either phi or psi is a constant function. Our results cover the minimal case (H=0), the case H^2 not equal to 1, and the critical case H^2=1, thereby extending the existing classification of parabolic surfaces in hyperbolic space.

What carries the argument

The product relation z = phi(x)psi(y) in the upper half-space model, which is substituted into the mean-curvature equation to obtain an ODE system that forces one factor constant.

Load-bearing premise

The surfaces are assumed to take exactly the product form z equals phi of x times psi of y with smooth functions phi and psi.

What would settle it

A pair of non-constant smooth functions phi and psi for which the mean curvature of the graph z equals phi of x times psi of y is constant would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.11649 by Rafael Belli, Rafael L\'opez.

**Figure 1.** Figure 1: Examples of homothetical surfaces of H3 with constant mean curvature H: from left to right, H = 0, H = − 1 2 , H = −1 and H = 2. 2. Preliminares In this section, we derive a suitable expression for the mean curvature of a homothetical surface. It is well known that the mean curvature H of a surface in H3 when the upper halfspace model is adopted, is related to its Euclidean mean curvature He by the formul… view at source ↗

read the original abstract

We classify all homothetical surfaces with constant mean curvature $H$ in the hyperbolic space $\mathbb{H}^3$. Using the upper half-space model with standard coordinates $(x,y,z)$, these surfaces are defined by the relation $z = \phi(x)\psi(y)$, where $\phi$ and $\psi$ are smooth functions of one variable. We demonstrate that any such surface is necessarily parabolic, meaning that either $\phi$ or $\psi$ is a constant function. Our results cover the minimal case ($H=0$), the case $H^2 \neq 1$, and the critical case $H^2=1$, thereby extending the existing classification of parabolic surfaces in hyperbolic space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies homothetical CMC surfaces in H^3 as necessarily parabolic by solving the separable PDE from the product graph ansatz across all H cases.

read the letter

The punchline is that any homothetical surface with constant mean curvature in hyperbolic 3-space must be parabolic. The authors prove that for graphs z = φ(x)ψ(y) in the upper half-space model, the CMC condition forces one of the functions to be constant, and they do this for H = 0, H² ≠ 1, and the critical H² = 1. They do this by computing the mean curvature explicitly, arriving at a separable equation, and then solving the resulting ODEs in each regime. The stress-test confirms the reductions are direct and free of unjustified steps. This extends previous work on parabolic CMC surfaces by showing the homothetical ones reduce to them. The paper does well on the case analysis, particularly the critical case where the equation changes character. The derivations are written out clearly enough that one can follow the algebra. A soft spot is the restriction to this specific separable form from the start. While that's the definition of homothetical here, it means the result doesn't say anything about whether there are other CMC surfaces that aren't of this product type. Also, the abstract claims a demonstration, but without seeing every line of the proof in the full text, one has to trust the case-by-case verification holds for all smooth φ, ψ. The citation pattern seems standard for the subfield, building on known results for parabolic surfaces. This paper is aimed at researchers in differential geometry who study surfaces in hyperbolic space or constant mean curvature problems. A reader interested in explicit classifications or examples in H^3 would find the complete list of such surfaces useful. It deserves serious refereeing because the central claim is a clean extension and the methods are standard but applied carefully to the critical case. I recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript classifies all homothetical surfaces with constant mean curvature H in hyperbolic 3-space H^3. In the upper half-space model these surfaces are graphs of the form z = φ(x)ψ(y) with smooth real functions φ and ψ. The central claim is that every such surface is necessarily parabolic: at least one of φ or ψ must be constant. The argument proceeds by deriving the CMC condition as a separable nonlinear PDE and then performing case analysis on the value of H, covering the minimal case H = 0, the regime H² ≠ 1, and the critical case H² = 1.

Significance. If the derivations hold, the result supplies a complete classification for this product-form class of CMC surfaces in H^3 and extends prior work on parabolic surfaces. The explicit computation of the first and second fundamental forms, the reduction to a separable PDE, and the exhaustive case analysis on H constitute the main technical contribution.

minor comments (2)

[Abstract] The abstract states that the work 'extends the existing classification of parabolic surfaces' but does not cite the specific references being extended; adding these citations would clarify the novelty.
[§4 (critical case)] In the critical case H² = 1 the final reduction to constant factors is stated without an explicit verification that the resulting ODE system admits no additional smooth non-constant solutions; a short remark confirming this would strengthen readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct PDE case analysis

full rationale

The manuscript begins from the explicit product ansatz z = φ(x)ψ(y) and the standard hyperbolic metric in the upper half-space model. It computes the first and second fundamental forms, inserts them into the mean-curvature formula, and obtains a separable nonlinear PDE. Case analysis on the constant H (including the critical H² = 1 regime) then shows algebraically that at least one factor must be constant. All reductions are explicit algebraic identities performed on the derived PDE; no parameter is fitted to data and then re-labeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The classification therefore rests on direct manipulation of the CMC condition rather than on any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard differential geometry assumptions for surfaces in hyperbolic space and the specific homothetical product ansatz.

axioms (2)

standard math φ and ψ are smooth functions of one variable.
Invoked in the definition of homothetical surfaces in the upper half-space model.
domain assumption The upper half-space model with standard coordinates represents hyperbolic 3-space.
Used to define the surfaces and compute mean curvature.

pith-pipeline@v0.9.0 · 5408 in / 1012 out tokens · 41965 ms · 2026-05-13T01:11:30.821844+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Theorem 1.2: If S has constant mean curvature, then either ϕ or ψ is a constant function. Proof proceeds by cases on H after obtaining the separable PDE (4) and applying Vieta + Lemma 2.1.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Equation (2) H = z H_e + N_3 and reduction to polynomial in e^{-w} with coefficients A0,A2,…

Reference graph

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16 extracted references · 16 canonical work pages

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