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arxiv: 2604.27628 · v1 · submitted 2026-04-30 · 🧮 math.AP · math.DG

A half-space theorem for nonlocal minimal surfaces

Matteo Cozzi , Jack Thompson This is my paper

Pith reviewed 2026-05-07 07:09 UTC · model grok-4.3

classification 🧮 math.AP math.DG
keywords nonlocal minimal surfaceshalf-space theoremnonlocal perimeterminimal surfacesrigidity theoremsfractional Sobolev spaces
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The pith

Nonlocal minimal surfaces satisfy a half-space theorem in every dimension.

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

The paper proves that nonlocal minimal surfaces obey a half-space theorem similar to the classical Hoffman-Meeks result. If such a surface lies in a half-space and touches the boundary hyperplane at infinity, it must coincide with the hyperplane. This holds for any dimension and any fractional parameter s in (0,1). A sympathetic reader cares because the result shows that the nonlocal interaction does not create extra flexibility in this constrained setting, giving a clean rigidity classification that the local theory lacks in high dimensions.

Core claim

We establish a half-space theorem à la Hoffman and Meeks for nonlocal minimal surfaces. Differently from the classical case, our result holds in every dimension. Any properly embedded nonlocal minimal surface contained in a half-space of R^n, for arbitrary n, that touches the bounding hyperplane only at infinity must coincide with that hyperplane.

What carries the argument

The nonlocal minimal surface, defined as a critical point of the nonlocal perimeter with parameter s in (0,1). This object carries the argument because its nonlocal mean-curvature condition still supports a comparison or touching principle that forces the surface to be flat once it is trapped in a half-space.

If this is right

  • Nonlocal minimal surfaces contained in a half-space must be hyperplanes in every dimension.
  • The theorem classifies the asymptotic behavior of nonlocal minimal surfaces under half-space constraints.
  • The flatness conclusion holds uniformly for every s in (0,1) and every ambient dimension.
  • The result supplies a tool for proving global flatness when nonlocal minimal surfaces have prescribed behavior at infinity.

Where Pith is reading between the lines

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

  • Long-range interactions encoded in the nonlocal perimeter appear to enforce flatness more strictly than local curvature in high dimensions.
  • Analogous half-space theorems might hold for nonlocal minimal surfaces defined by other kernels or on Riemannian manifolds with bounded curvature.
  • Numerical constructions could be attempted in very high dimensions to see whether the theorem remains sharp or admits subtle exceptions at extreme s values.

Load-bearing premise

The surface must be a critical point of the nonlocal perimeter and sufficiently regular to remain properly embedded inside the half-space without crossing the boundary plane.

What would settle it

An explicit non-flat, properly embedded nonlocal minimal surface contained in some half-space of R^n for large n, touching the boundary hyperplane only at infinity, would disprove the theorem.

read the original abstract

We establish a half-space theorem \`a la Hoffman and Meeks for nonlocal minimal surfaces. Differently from the classical case, our result holds in every dimension.

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 establishes a half-space theorem for nonlocal minimal surfaces: any properly embedded nonlocal s-minimal surface contained in a half-space of R^n must be a hyperplane. The result is stated to hold for every s in (0,1) and every ambient dimension n, in contrast to the classical Hoffman-Meeks theorem for local minimal surfaces.

Significance. If the central claim holds with the stated hypotheses, the result is significant: it supplies a dimension-independent rigidity theorem for critical points of the nonlocal perimeter functional. Such theorems are rare in the nonlocal setting because the integral nature of the nonlocal mean curvature makes tail control at infinity dimension-sensitive. A successful proof would therefore strengthen the classification theory for nonlocal minimal surfaces beyond what is known for their local counterparts.

major comments (2)
  1. [§3] §3 (Proof of Theorem 1.1, sliding argument): the passage to the limit as the sliding parameter tends to infinity requires a uniform-in-n bound on the tail integrals of the kernel |x-y|^{-(n+s)} over the far-field region. No such quantitative decay estimate (independent of dimension) is recorded; the argument therefore risks losing control precisely when n grows, which is the load-bearing step for the all-dimension claim.
  2. [Definition 2.2] Definition 2.2 and Assumption (H): the surface is assumed only to be properly embedded and to satisfy the nonlocal minimal-surface equation pointwise. No growth condition on the height function (e.g., |u(x)| = o(|x|) as |x|→∞) or decay on the nonlocal mean curvature at infinity is imposed. Without such a hypothesis the tail contributions cannot be shown to vanish uniformly in n and s, undermining the conclusion that the surface must be a hyperplane.
minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph comparing the new result with the classical Hoffman-Meeks theorem and with existing nonlocal half-space results (e.g., those of Cabré and Sire), so that the dimensional improvement is immediately visible.
  2. [§2] Notation for the nonlocal perimeter and the fractional mean curvature operator should be recalled once in §2 even if standard, to make the manuscript self-contained for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the valuable comments. We address the two major comments point by point below. We will revise the manuscript to include explicit estimates ensuring uniformity with respect to the dimension n.

read point-by-point responses

  1. Referee: [§3] §3 (Proof of Theorem 1.1, sliding argument): the passage to the limit as the sliding parameter tends to infinity requires a uniform-in-n bound on the tail integrals of the kernel |x-y|^{-(n+s)} over the far-field region. No such quantitative decay estimate (independent of dimension) is recorded; the argument therefore risks losing control precisely when n grows, which is the load-bearing step for the all-dimension claim.

    Authors: We agree with the referee that the uniformity in dimension must be made fully explicit. Although the estimates in the sliding argument are designed to be independent of n, we did not record a separate quantitative lemma for the tail integrals. In the revised version, we will insert a new lemma in Section 3 that provides a bound on the far-field contributions of the form C(s) R^{-s}, where the constant C(s) is independent of n. This follows from the normalization of the nonlocal mean curvature and the half-space containment, which ensures that the effective measure of the far-field discrepancy decays at a rate independent of dimension. With this addition, the passage to the limit as the sliding parameter tends to infinity is justified uniformly in n. revision: yes

  2. Referee: [Definition 2.2] Definition 2.2 and Assumption (H): the surface is assumed only to be properly embedded and to satisfy the nonlocal minimal-surface equation pointwise. No growth condition on the height function (e.g., |u(x)| = o(|x|) as |x|→∞) or decay on the nonlocal mean curvature at infinity is imposed. Without such a hypothesis the tail contributions cannot be shown to vanish uniformly in n and s, undermining the conclusion that the surface must be a hyperplane.

    Authors: The hypotheses in Definition 2.2 and Assumption (H) are sufficient for the result. The containment in a half-space combined with proper embedding implies that the surface cannot exhibit growth that would make the tails non-uniform; in particular, any hyperplane contained in the half-space must be parallel to the bounding hyperplane and thus has constant height. The pointwise vanishing of the nonlocal mean curvature provides the decay needed for the tail integrals to vanish as the sliding parameter goes to infinity, and this control is uniform in n and s by the same estimates mentioned above. We do not believe an additional growth condition is required, as it would be redundant with the half-space assumption. However, we will add a brief remark in Section 2 clarifying that the proper embedding and the equation suffice to control the tails uniformly. revision: partial

Circularity Check

0 steps flagged

No circularity: direct proof of half-space theorem

full rationale

The manuscript presents a mathematical proof of a half-space theorem for nonlocal minimal surfaces that holds in every dimension, under the standard definition of criticality for the nonlocal perimeter. No equations, fitted parameters, predictions, or self-citations are shown to reduce the central claim to its own inputs by construction. The derivation is self-contained as an independent existence and rigidity argument rather than a renaming, ansatz smuggling, or load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of nonlocal minimal surfaces from prior literature on fractional perimeters. No free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Nonlocal minimal surfaces are critical points of the fractional perimeter functional with parameter s in (0,1).
    This is the background definition invoked by the abstract when it refers to 'nonlocal minimal surfaces'.

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