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arxiv: 2605.20964 · v1 · pith:C47BOED7new · submitted 2026-05-20 · 🧮 math.DG

Regularity of stable capillary minimal hypersurfaces

Gaoming Wang , Xuwen Zhang This is my paper

Pith reviewed 2026-05-21 02:11 UTC · model grok-4.3

classification 🧮 math.DG
keywords capillary minimal hypersurfacesstability inequalityintegral curvature estimatesregularity theoryBernstein theoremhalf-spaceminimal conesboundary sheeting
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The pith

Stable capillary minimal hypersurfaces in half-spaces satisfy regularity and compactness that yield a generalized Bernstein theorem for complete embedded examples with Euclidean area growth.

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

This paper develops a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space with fixed contact angle between 0 and pi and dimension at least 2. The central result is a generalized Bernstein theorem asserting that any embedded complete stable capillary minimal hypersurface with Euclidean area growth must be flat. The proof begins with an integral curvature estimate obtained by choosing a tilt excess function that removes all boundary terms from the stability inequality. This estimate supports a boundary sheeting theorem and a refined classification of stable capillary minimal cones, which together produce the full regularity and compactness statements.

Core claim

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space H^{n+1} with contact angle theta in (0,pi) and dimension n at least 2. The key step is an integral curvature estimate produced by selecting an appropriate tilt excess function that eliminates boundary terms in the stability inequality. Building on this estimate, a boundary sheeting theorem is established by refining earlier arguments, and a refined classification of stable capillary minimal cones is obtained. These ingredients together deliver the regularity and compactness theorems, and as a consequence any embedded complete stable capillary minimal hypersurface in H^{n+1} with only a

What carries the argument

An integral curvature estimate for stable capillary minimal hypersurfaces, obtained by choosing a tilt excess function that cancels boundary terms in the stability inequality.

If this is right

  • Stable capillary minimal hypersurfaces are regular away from the boundary.
  • Compactness holds for bounded sequences of such hypersurfaces.
  • Stable capillary minimal cones admit a refined classification.
  • The generalized Bernstein theorem applies precisely to the complete embedded case with Euclidean area growth.

Where Pith is reading between the lines

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

  • The technique of cancelling boundary terms with a tilt excess function could be tested on capillary problems in domains other than the half-space.
  • The resulting regularity might be combined with variational methods to produce existence results for stable capillary surfaces with prescribed contact angle.
  • Similar estimates may apply when the contact angle varies mildly along the boundary rather than remaining constant.

Load-bearing premise

The stability inequality for capillary minimal hypersurfaces admits a tilt excess function that removes every boundary term while still producing a usable positive curvature integral.

What would settle it

An explicit example of a non-flat complete embedded stable capillary minimal hypersurface in the half-space that has only Euclidean area growth would show the generalized Bernstein theorem is false.

Figures

Figures reproduced from arXiv: 2605.20964 by Gaoming Wang, Xuwen Zhang.

Figure 1
Figure 1. Figure 1: The case when γ ∈ Γ1 Case 1.1: i1 ̸= i2 and at least one of i1, i2 is not 0 or π. By construction and the definition of θ, the angle between νk,i1,j1 (X1) and νk,i2,j2 (X2) is bounded below by min |θi1 − θi2 |, π − |θi1 − θi2 | [PITH_FULL_IMAGE:figures/full_fig_p048_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The case when γ ∈ Γ2 and C is not in Cθ νk Hθ Hπ−θ γ Mk,2,j X2 νk X1 Sτ x1 xn+1 [PITH_FULL_IMAGE:figures/full_fig_p049_2.png] view at source ↗
read the original abstract

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. As a consequence, we obtain the generalized Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in $\mathbb{H}^{n+1}$ with Euclidean area growth. The key innovation is an integral curvature estimate: by carefully selecting an appropriate tilt excess function, we are able to eliminate the boundary terms arising in the stability inequality. Building on this, we establish a boundary sheeting theorem by refining the arguments in [SS81]. These results, combined with a refined classification of stable capillary minimal cones, lead to the main regularity and compactness theorems.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space H^{n+1} with fixed contact angle θ ∈ (0,π) and n ≥ 2. The central technical step is an integral curvature estimate obtained by substituting a tilt excess function into the stability inequality so that all boundary terms vanish. Building on this, the authors establish a boundary sheeting theorem by refining the arguments of [SS81], obtain a refined classification of stable capillary minimal cones, and deduce regularity, compactness, and a generalized Bernstein theorem for complete embedded stable capillary minimal hypersurfaces with Euclidean area growth.

Significance. If the estimates hold, the work provides a meaningful extension of Bernstein-type theorems to the capillary setting with general contact angles. The approach of engineering a tilt excess to cancel boundary contributions in the stability inequality is a concrete technical contribution that may apply to other free-boundary or capillary problems in geometric analysis.

major comments (2)
  1. [integral curvature estimate section] The integral curvature estimate (developed in the section following the stability inequality) rests on the assertion that a specific tilt excess function eliminates every boundary integral for arbitrary θ ∈ (0,π). The manuscript should supply the explicit computation of the boundary terms after substitution, including the precise cancellation identity and verification that no residual term survives under the Euclidean area growth hypothesis.
  2. [cone classification section] The refined classification of stable capillary minimal cones (used to close the regularity argument) is invoked to rule out non-flat cones at the boundary. The manuscript should state explicitly which additional cases arise for θ ≠ π/2 and confirm that the stability inequality plus the new curvature estimate suffice to exclude them.
minor comments (2)
  1. [preliminaries] The notation for the tilt excess function and the precise definition of Euclidean area growth should be introduced with a displayed equation before they are used in the stability manipulation.
  2. [introduction] A short comparison paragraph with the classical Bernstein theorem for minimal hypersurfaces (without boundary) would help readers gauge the new difficulties introduced by the capillary condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We address the major comments point by point below and will revise the paper to incorporate the requested explicit details.

read point-by-point responses

  1. Referee: [integral curvature estimate section] The integral curvature estimate (developed in the section following the stability inequality) rests on the assertion that a specific tilt excess function eliminates every boundary integral for arbitrary θ ∈ (0,π). The manuscript should supply the explicit computation of the boundary terms after substitution, including the precise cancellation identity and verification that no residual term survives under the Euclidean area growth hypothesis.

    Authors: We agree that an explicit verification of the boundary cancellation would improve the presentation. While the manuscript states that the chosen tilt excess function eliminates all boundary integrals for general θ ∈ (0,π), the detailed computation of each term and the resulting cancellation identity were not written out in full. In the revised manuscript we will insert the complete calculation immediately after the stability inequality, showing term-by-term that the boundary contributions cancel, and we will verify that the Euclidean area growth hypothesis leaves no residual boundary term. This addition will be placed in the integral curvature estimate section. revision: yes

  2. Referee: [cone classification section] The refined classification of stable capillary minimal cones (used to close the regularity argument) is invoked to rule out non-flat cones at the boundary. The manuscript should state explicitly which additional cases arise for θ ≠ π/2 and confirm that the stability inequality plus the new curvature estimate suffice to exclude them.

    Authors: We accept the referee’s suggestion to make the classification more explicit for general contact angles. For θ ≠ π/2 the possible non-flat stable capillary cones include certain tilted or angled cones that do not appear when θ = π/2. In the revised version we will list these additional cases in the cone classification section and supply a short argument showing that each is ruled out by combining the stability inequality with the integral curvature estimate already established. This will close the regularity argument without altering the overall logic. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses new estimates and external refinements

full rationale

The paper's chain proceeds from the stability inequality to an integral curvature estimate via a chosen tilt excess function that cancels boundary terms, followed by a boundary sheeting theorem refining [SS81] and a cone classification. No quoted step reduces the main result to a fitted input, self-definition, or self-citation loop; the estimates are presented as novel technical work building on external references without load-bearing self-citations or ansatz smuggling. The generalized Bernstein theorem follows from these independent steps under the stated area growth hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard differential geometry background (stability inequality for capillary surfaces, regularity theory for minimal hypersurfaces) plus the new technical choices; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The stability inequality holds for capillary minimal hypersurfaces with fixed contact angle in the half-space.
    Invoked as the starting point for the integral curvature estimate.
  • domain assumption Standard regularity and compactness results from minimal surface theory extend after the boundary terms are removed.
    Used to pass from the curvature estimate to the sheeting theorem and regularity.

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    work page doi:10.48550/arxiv.2410.20548 2024