arxiv: 2605.11646 · v1 · submitted 2026-05-12 · 🧮 math.DG

Recognition: no theorem link

Circles-foliated stationary surfaces of the Dirichlet energy

Rafael L\'opez

Pith reviewed 2026-05-13 01:16 UTC · model grok-4.3

classification 🧮 math.DG

keywords anisotropic mean curvatureDirichlet energycircle foliationRiemann examplesminimal surfacesstationary surfaceshorizontal planes

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The pith

Non-rotational surfaces foliated by horizontal circles achieve zero anisotropic mean curvature for the Dirichlet energy.

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

The paper proves the existence of surfaces that are stationary for the Dirichlet energy with zero anisotropic mean curvature and are foliated by circles lying in parallel horizontal planes, but are not rotationally symmetric around the vertical axis. These surfaces are described geometrically and extend the classical Riemann examples from the theory of minimal surfaces to this anisotropic variational problem. Additionally, a classification is given showing that any circle-foliated surface with zero anisotropic mean curvature is either axially symmetric or one of these new examples. A general reader might care because such surfaces arise in variational problems where the energy depends on the gradient in a non-isotropic way, as in certain physical models.

Core claim

We prove the existence of non-rotational surfaces with Λ=0 and foliated by a one-parameter family of circles contained in horizontal planes obtaining a geometric description of them. These surfaces extend the known Riemann examples of the theory of minimal surfaces to the anisotropic context of the Dirichlet energy. We classify all surfaces with zero anisotropic mean curvature foliated by circles proving that either the surface is axially symmetric about the z-axis or the surface belongs to one of the above examples.

What carries the argument

Foliation by a one-parameter family of circles contained in horizontal planes, which reduces the anisotropic mean curvature condition to a geometric description or solvable equation.

If this is right

These surfaces provide explicit non-symmetric examples of stationary surfaces for the Dirichlet energy with zero anisotropic mean curvature.
The classification shows that no other circle-foliated surfaces with zero anisotropic mean curvature exist beyond the axially symmetric ones and the new family.
The construction and classification extend in part to the case of nonzero constant anisotropic mean curvature.
The results link classical minimal surface theory to anisotropic variational problems through concrete geometric examples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The geometric description may yield explicit parametrizations that permit direct stability analysis or numerical visualization of the surfaces.
Foliation assumptions of this type could produce analogous classifications for other anisotropic energies or curvature functionals.
The examples might serve as benchmarks for testing numerical solvers in anisotropic geometric variational problems.

Load-bearing premise

The surfaces are sufficiently smooth and exactly foliated by a regular one-parameter family of circles lying in distinct horizontal planes.

What would settle it

Construct the surface from the geometric description in the paper and compute its anisotropic mean curvature directly to check if it vanishes identically, or exhibit a circle-foliated surface with zero anisotropic mean curvature that is neither axially symmetric nor one of the constructed examples.

Figures

Figures reproduced from arXiv: 2605.11646 by Rafael L\'opez.

**Figure 1.** Figure 1: Surfaces Σλ,c and Σeλ,c of type I. Here λ = 2 and c = 1. Left: the surface Σλ,c, (19). Right: the surface Σeλ,c after vertical translations of Σλ,c. Horizontal (black) straight-lines are contained on the surface. Theorem 3.3 (Type II). Let Σλ,c be a surface parametrized by (20) X(s, θ) = −1 + cos θ λs + c , sin θ λs + c , s , s ∈ I := (− c λ , ∞), θ ∈ R. Here a(s) = − 1 λs + c , r(s) = 1 λs + c , c > 0.… view at source ↗

**Figure 2.** Figure 2: Surfaces Σλ,c and Σeλ,c of type II. Here λ = 1 and c = 0. Left: the surface Σλ,c, (20). Right: the surface Σeλ,c after the 1800 rotation about L of Σλ,c. Remark 3.4. Notice that the surface (20) contains the z-axis, which it is a vertical line. However, we cannot assert that Σλ,c may be extended by reflections across the z-axis because the coordinate function y = y(x, z) on the surface is not harmonic in g… view at source ↗

**Figure 3.** Figure 3: Surfaces Σλ,c and Σeλ,c of type III. Here λ = 1 and c = 0. Left: the surface Σλ,c, (21). Right: the surface Σeλ,c after the 1800 rotation about L of Σλ,c. In [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Cross-section of the surfaces (19) (left), (20) (middle) and (21) (right) with the plane y = 0. The dashed line is the curve of centers c(s) of the circles. The blue lines represents the intersection of the surface with the plane y = 0. 4. Proof of Theorem 1.3 The separate the proof of Thm. 1.3 in two subsections. We first prove that the planes of the foliation must be parallel. Next, we will show that the… view at source ↗

read the original abstract

In Euclidean space we study surfaces with constant anisotropic mean curvature $\Lambda$ of the Dirichlet energy $\int_\Omega( |Du|^2+\Lambda u)$. We prove the existence of non-rotational surfaces with $\Lambda=0$ and foliated by a one-parameter family of circles contained in horizontal planes obtaining a geometric description of them. These surfaces extend the known Riemann examples of the theory of minimal surfaces to the anisotropic context of the Dirichlet energy. More general, we classify all surfaces with zero anisotropic mean curvature foliated by circles proving that either the surface is axially symmetric about the $z$-axis or the surface belongs to one of the above examples. We also study the case that the anisotropic mean curvature is a non-zero constant.

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 circle-foliated surfaces with zero anisotropic mean curvature for the Dirichlet energy in R^3 and adds explicit non-rotational examples that extend the classical Riemann surfaces.

read the letter

The paper classifies all surfaces in Euclidean 3-space foliated by circles in horizontal planes that have zero anisotropic mean curvature with respect to the Dirichlet energy. It also constructs a one-parameter family of non-rotational examples and shows they complete the list together with the axially symmetric case. The reduction of the curvature condition to an ODE on the center trajectory and radius function is the core step, and the solutions are described geometrically. This extends the known Riemann examples without forcing rotational symmetry. The zero-curvature case is handled in detail, while the constant non-zero case receives only partial treatment. The arguments rely on standard smoothness and regularity assumptions for the foliation, which are typical in this area and do not introduce circularity. The ODE analysis appears direct and covers the regular cases without visible internal contradictions. One clear limitation is the narrow setting: everything is restricted to horizontal planes in R^3, so the results do not immediately apply to tilted foliations or higher dimensions. The proofs are ODE-based rather than using new variational techniques, which keeps them verifiable but limits broader impact. Readers working on anisotropic minimal surfaces or foliated variational problems will find the explicit examples and classification useful. A specialist in geometric analysis who already knows the classical Riemann surfaces can extract the extension quickly. The work is focused enough that it deserves a serious referee. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies surfaces in Euclidean 3-space with constant anisotropic mean curvature Λ associated to the Dirichlet energy functional ∫(|Du|² + Λu). It proves existence of non-rotational surfaces with Λ = 0 that are foliated by a one-parameter family of circles lying in horizontal planes, supplies a geometric description of these surfaces, and shows they extend the classical Riemann examples from minimal surface theory to the anisotropic setting. It further classifies all zero-anisotropic-mean-curvature surfaces foliated by circles, proving that any such surface is either axially symmetric about the z-axis or belongs to the newly constructed family. The paper also treats the case of non-zero constant anisotropic mean curvature.

Significance. If the derivations hold, the work supplies the first explicit non-rotational examples of anisotropic minimal surfaces (Λ = 0) under a natural foliation hypothesis and gives a complete classification within that class. The reduction of the anisotropic mean-curvature equation to an ODE system on the center curve c(h) and radius function r(h) is constructive and directly generalizes the classical Riemann construction; this offers a concrete tool for further study of anisotropic variational problems and may serve as a benchmark for numerical or stability analyses.

minor comments (3)

[Abstract] The abstract refers to “the above examples” without having defined them within the abstract; rephrase the final sentence of the abstract so that it is self-contained.
[§1 or Introduction] The precise relation between the anisotropic mean curvature and the Poisson equation Δu = Λ/2 (mentioned in the reader’s summary) should be stated explicitly in the introduction or in the first section where the energy is introduced, including the precise definition of the anisotropic mean curvature operator.
[Classification theorem (likely §4 or §5)] In the classification statement, clarify whether the foliation is assumed to be regular (non-vanishing gradient, smooth dependence on the height parameter) throughout the surface or only away from possible singular loci; this assumption is load-bearing for the ODE reduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The referee's summary and significance statement accurately reflect the main results: the construction of non-rotational circle-foliated surfaces with zero anisotropic mean curvature for the Dirichlet energy, their geometric description as extensions of the classical Riemann examples, the complete classification within the circle-foliated class, and the treatment of the constant non-zero case. We are pleased that the constructive ODE reduction on the center curve and radius function is viewed as a useful tool for further study. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The paper assumes surfaces foliated by a regular one-parameter family of circles in horizontal planes and derives the zero anisotropic mean curvature condition from the variational problem for the Dirichlet energy. This reduces to an ODE system on the center trajectory c(h) and radius r(h), whose solutions are classified into axially symmetric cases or a one-parameter family of non-rotational examples. The classification and existence claims follow from standard geometric analysis applied to the foliation assumption and the Euler-Lagrange equation, without any reduction of outputs to fitted parameters, self-definitional loops, or load-bearing self-citations. The extension of Riemann examples is a geometric description derived independently rather than by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from differential geometry and variational calculus; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption The surfaces under consideration are sufficiently smooth (at least C^2) for the anisotropic mean curvature to be well-defined.
Invoked implicitly when stating the constant anisotropic mean curvature condition and the foliation.
domain assumption The foliation consists of a regular one-parameter family of circles lying in distinct horizontal planes.
This is the structural hypothesis on which both the existence construction and the classification theorem are based.

pith-pipeline@v0.9.0 · 5409 in / 1497 out tokens · 49061 ms · 2026-05-13T01:16:12.089740+00:00 · methodology

discussion (0)

Reference graph

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