Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

Michele Caselli

REVIEW 3 major objections 4 minor 27 references

Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For the fractional perimeter with $s$ very close to 0, any stable cone in the plane is a half-plane.

desk verdict The intended classification may be true, but the proof stands on a false exponent in Lemma 3.1; the paper needs a major repair before it can be trusted. read the letter →

arxiv 2412.06318 v3 pith:5T33ZMXM submitted 2024-12-09 math.AP

Michele Caselli This is my paper · ORCID

classification math.AP MSC 35R1149Q20

keywords fractionalperimeters-minimalconesstabilityHardyinequalitynonlocalminimalsurfacesMorseindexclassificationofintheplane

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 establishes that, for the fractional (nonlocal) perimeter with exponent $s$ small enough, the only cones in $\mathbb{R}^2$ that are stationary and stable away from the origin are half-planes. The result is nonlocal in character: for the classical perimeter (formally $s=1$) and for $s$ close to 1, the cross-shaped cone $\{xy>0\}$ is stable in $\mathbb{R}^2\setminus\{0\}$. The paper only assumes stability away from the cone's vertex, so the theorem does not use any stability information at the singular point itself. It also derives the same classification for $s$-minimal cones with finite Morse index.

What carries the argument

The engine is Hardy's inequality for the $H^\sigma(\mathbb{R})$ seminorm with $\sigma=(1+s)/2$, whose optimal constant is of order $s^2$ as $s\to0$, together with radial functions in $C_c^2(\mathbb{R}\setminus\{0\})$ that nearly saturate it. Plugging such a radial function into the second variation inequality for the cone separates the problem into a Hardy part on each ray and an angular sum; the coefficient $1/s^2$ from saturation competes with the number of rays, forcing the cone to have exactly one sector. The auxiliary machinery consists of the BV-estimate for stable $s$-minimal surfaces with constant $C/s$, the representation of nonlocal mean curvature as a boundary integral, and the finite-index-to-stability lemma that yields the corollary.

What would settle it

Evaluate the integral $\int_0^\infty dt/(1+t^2-2t\cos\theta)^{(2+s)/2}$ numerically for a small $s$ and a sequence of small angles $\theta$, comparing its growth as $\theta\to0$ with the two candidate powers of $1-\cos\theta$; the result settles whether Lemma 3.1, and with it the proof of Theorem 1.3, is valid. A second check is to apply the radial Hardy-saturating test function to a two-ray cone with a very small angle and see whether the stability inequality fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.3: there exists $s_0\in(0,1/2)$ such that for every $s\in(0,s_0)$, every $s$-minimal cone $E\subset\mathbb{R}^2$ that is stable in $\mathbb{R}^2\setminus\{0\}$ in the sense of inner variations is a half-plane. The proof shows that a stable cone can have at most a controlled number of rays, then uses radially symmetric test functions that nearly saturate Hardy's inequality on the half-line to force $N=1$, and finally uses the first variation formula at a smooth boundary point to force the single angle to be $\pi$. Corollary 1.4 extends the conclusion to cones with finite Morse index.

Load-bearing premise

The proof depends on Lemma 3.1's lower bound for the interaction between two rays, which decays like $(1-\cos\theta)^{-(1+s)}$ in the angle $\theta$ between the rays; if the correct small-angle decay is the weaker $(1-\cos\theta)^{-(1+s)/2}$, the argument forcing a stable cone to have only two rays and then angle $\pi$ collapses.

Editorial extensions

If this is right

For small $s$, the fractional widths on a Riemannian surface are attained by smooth $s$-minimal surfaces: every blow-up cone is flat and the improvement-of-flatness theorem gives regularity.
The cross $\{xy>0\}$ is unstable in $\mathbb{R}^2\setminus\{0\}$ for small $s$, despite being stable for the classical perimeter and expected to be stable for $s$ close to 1.
The number of rays of a stable cone is bounded by a constant times $1/s$.
The classification extends from stability to finite Morse index.
There is a sharp contrast with higher dimensions: in $\mathbb{R}^7$, smooth stable non-flat $s$-minimal cones exist for small $s$.

Reading between the lines

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

A natural stress test is to replace Lemma 3.1's angular lower bound with the true small-angle scaling and see whether the Hardy-saturation argument still forces $N=1$; if it does, the classification survives with a different quantitative range for $s_0$.
The same radial Hardy-saturation mechanism could be tried in higher dimensions, where it might separate radial from angular effects for other conical singularities and yield new flatness results for stable $s$-minimal cones.
The instability of the cross for small $s$ suggests a phase transition as $s$ varies, so locating the threshold would connect this small-$s$ classification to the known stability of crosses near $s=1$.

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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: interesting question, fresh mechanism, but Lemma 3.1 is wrong, and it carries the whole proof of Theorem 1.3. I side with the stress-test note.

What's new: Theorem 1.3, if true, would be the first classification of stable s-minimal cones in the punctured plane for small s, contrasting with s near 1 and the classical perimeter. The Hardy-saturation idea for s→0 is genuinely new, and the BV-estimate for small s (Theorem 2.8) looks like a useful standalone result. The appendix proof that the cross is unstable for small s is independent and appears sound.

The soft spot is not minor. In Lemma 3.1, the claimed bound is off by a factor (1-cosθ)^{-(1+s)/2}. The integral over the ray is asymptotic to c/(|x|^{1+s}(1-cosθ)^{(1+s)/2}); the substitution in the proof pulls out the wrong power and then renormalizes the t² term incorrectly. With the correct exponent, equation (8) no longer forces the blow-up used in the Claim, and the argument does not rule out many nearly parallel rays. The contradiction that yields N=1 evaporates. Everything after Lemma 3.1 depends on that false lower bound, so Theorem 1.3 is not proven. The finite-index extension in Section 4 inherits the failure.

That said, the paper is honest and competently written; the flaw is a technical error rather than fraud or confusion. The appendix and the BV estimate deserve to survive independently.

Bottom line: this deserves a serious referee, but not publication as is. I would send it to review with the expectation that the author must either fix Lemma 3.1 or develop a different mechanism. If the correct exponent still allows a similar argument, the result might be salvageable; as written, it does not.

Referee Report

3 major / 4 minor

Summary. The paper claims that for sufficiently small s in (0,1), the only s-minimal cones in R^2 that are stable in R^2 \ {0} are half-planes (Theorem 1.3), and it derives from this a classification of finite-Morse-index cones (Corollary 1.4). The strategy is to use a BV-estimate for small s to show the cone has finitely many rays, then to use the second variation formula with a radial test function that nearly saturates Hardy's inequality on (0,∞). A key quantitative lemma (Lemma 3.1) is used to force a contradiction unless the cone has exactly one sector of angle π. The appendix gives a separate proof that the cross {xy>0} is unstable for small s.

Significance. If Theorem 1.3 were correct, it would be a striking purely nonlocal phenomenon: it would contradict the classical-perimeter behavior and the expected behavior for s close to 1, and it would imply regularity of the fractional min-max surfaces discussed in Section 1.1. The paper also contains useful auxiliary ingredients, notably Lemma 2.6 on the boundary-integral representation of the nonlocal mean curvature for non-smooth sets, a BV-estimate with explicit dependence on s as s→0, and an independent instability proof for the cross in the Appendix. These components appear to be of independent interest. However, the central classification rests on Lemma 3.1, and that lemma is false; the main theorem is therefore not established.

major comments (3)

[Section 3, Lemma 3.1] The claimed lower bound is false. Let a = 1 - cos(θ) and write the integral in the lemma as |x|^{-(1+s)} I(a), with I(a) = ∫_0^∞ dt ((t-1)^2 + 2a t)^{-(2+s)/2}. For small a the dominant contribution comes from t near 1; setting u = t - 1 gives I(a) ~ a^{-(1+s)/2} ∫_R (u^2 + 2)^{-(2+s)/2} du. The correct growth is therefore a^{-(1+s)/2}, not a^{-(1+s)}. The step in the proof that replaces (t^2 + 3a)^{-(2+s)/2} by a^{-(1+s)} (t^2 + 3)^{-(2+s)/2} is invalid, because the integral over the t-variable has a concentration window of width √a, which changes the exponent by (1+s)/2 instead of 1+s. Consequently the factor (1 - cos(θ))^{-(1+s)} in equation (8) is not available.
[Section 3, proof of Theorem 1.3 after equation (8)] Even if Lemma 3.1 were replaced by the correct asymptotic, the final contradiction forcing N = 1 does not follow. With the correct exponent, the term for i = j + 2 in the displayed sum becomes (1 - cos(θ_j^{j+2}))/(1 - cos(θ_j^{j+2}))^{(1+s)/2} = (1 - cos(θ_j^{j+2}))^{(1-s)/2}, which tends to 0 for nearly parallel rays. Thus the inequality ≤ 1/100 is no longer impossible, and the argument does not exclude cones with N ≥ 2 rays. Since Lemma 3.1 is the only estimate producing the critical divergence in (8), the proof of Theorem 1.3 collapses. Corollary 1.4 and the regularity discussion in Section 1.1 inherit this gap.
[Section 3, Claim (9)] There is a further algebraic issue in the derivation of the contradiction from (7). The inequalities s^2 ≥ c/(100N) and N ≤ C/s imply s ≥ c/(50C), not that s is small; the contradiction holds only for s below that fixed constant, which can be absorbed in s_0. This point is fixable and is not the main obstruction, but it should be corrected if the argument is revised.

minor comments (4)

[Section 3, equation (8)] The denominator in the display after the Hardy-saturation step is written as |x-y|^{1+s}, whereas the stability inequality and the preceding displays use |x-y|^{2+s}. This appears to be a typo, but it creates a dimensional mismatch and should be corrected.
[Section 3, proof of Lemma 3.1] The change of variables in the displayed chain of inequalities has reversed integration limits (∫_{1/2}^{-1/2} and ∫_{1/10}^{-1/10}); these should read ∫_{-1/2}^{1/2} and ∫_{-1/10}^{1/10}. Correcting the limits does not repair the exponent error noted above.
[Section 1, paragraph before Corollary 1.4] There is a typo in 'conclusialon'; it should be 'conclusion'.
[Section 3, notation] The definition of θ_j^i as 'the counterclockwise angle from Σ_i and Σ_j' is ambiguous; it should specify the angle from Σ_i to Σ_j in the chosen orientation, and the modulo-2π convention should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the s→0 classification is obtained from independent Hardy and stability inputs.

full rationale

The paper's main theorem (Theorem 1.3) is not circular. The inputs are the sharp Hardy inequality for the H^σ(R) seminorm (Theorem 2.1, cited to Frank–Seiringer), the second variation formula for the s-perimeter (Theorem 2.7), and a BV-estimate whose proof is adapted from Cinti–Serra–Valdinoci with explicitly tracked constants. The proof substitutes a radial test function saturating Hardy's inequality into the stability inequality; no quantity appearing in the conclusion—flatness of stable s-minimal cones for small s—is used to define the test function, the constants, or the claimed estimate. The author's prior works [CFSS23], [FS24], and [CFSS24] are cited for auxiliary regularity, Morse-index, and second-variation facts, but the central small-s classification does not reduce to those citations; the load-bearing estimate leading to Claim (9) is derived in the paper from the stated Hardy inequality and Lemma 3.1. Any potential issue with the asymptotic exponent in Lemma 3.1 is a mathematical correctness concern about a direct integral estimate, not a circularity: it does not make the conclusion an input by definition or by self-citation. Accordingly, no circular step can be exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proofs depend on external results: the sharp Hardy inequality with saturation (FS08), the second variation formula (FFM+15, DdPW18, FS24), and the BV estimate with 1/s scaling (CSV19, FS24, Tho24). No parameters are fitted to data; the constant s0 is existential. No new entities are introduced. The critical flaw is an incorrect integral estimate in Lemma 3.1, not circularity.

assumptions (4)

standard math Sharp Hardy inequality for H^sigma(R) with radial saturation (Theorem 2.1)
Taken from Frank-Seiringer (FS08); provides the test function with the 1/s^2 factor.
domain assumption BV estimate for stable s-minimal surfaces: Per(E, B_{1/2}) <= C/s as s to 0 (Theorem 2.8)
Used to bound the number of cone rays by C/s. The proof is sketched and depends on CSV19 with claimed constant tracking.
standard math Second variation formula for the s-perimeter (Theorem 2.7)
Source of the stability inequality (5). Cited from FFM+15, DdPW18, FS24.
domain assumption Finite-index-to-stability lemma for cones (Lemma 4.3, from CFSS23)
Used for Corollary 1.4; cited, not reproved.

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Cite this review

Pith. "Pith review of Stable $s$-minimal cones in $\mathbb{R}^2$ are flat for $s \sim 0$." pith.science (2026). https://pith.science/paper/5T33ZMXM

@misc{pith2026241206318,
  author       = {Pith},
  title        = {Pith review of: Stable $s$-minimal cones in $\mathbbR^2$ are flat for $s \sim 0$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5T33ZMXM}},
  note         = {Machine review of arXiv:2412.06318}
}

abstract

For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the regime $s$ close to $1$, where nontrivial cones as $\{xy>0\} \subset \mathbb{R}^2$ are stable for inner variations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Cabr\'e, E

X. Cabr\'e, E. Cinti, and J. Serra. Stable s -minimal cones in R ^3 are flat for s 1 . J. Reine Angew. Math. , 764:157--180, 2020

work page 2020

[2] [2]

Cabr \'e , E

X. Cabr \'e , E. Cinti, and J. Serra. Stable solutions to the fractional allen-cahn equation in the nonlocal perimeter regime. to appear on American Journal of Mathematics , 2021

work page 2021

[3] [3]

H. Chan, S. Dipierro, J. Serra, and E. Valdinoci. Nonlocal approximation of minimal surfaces: optimal estimates from stability. arXiv:2308.06328 , 2023

arXiv 2023

[4] [4]

Ciraolo, A

G. Ciraolo, A. Figalli, F. Maggi, and M. Novaga. Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. J. Reine Angew. Math. , 741:275--294, 2018

work page 2018

[5] [5]

Cabr\'e, M

X. Cabr\'e, M. M. Fall, J. Sol\`a-Morales, and T. Weth. Curves and surfaces with constant nonlocal mean curvature: meeting A lexandrov and D elaunay. J. Reine Angew. Math. , 745:253--280, 2018

work page 2018

[6] [6]

Caselli, E

M. Caselli, E. Florit-Simon, and J. Serra. Yau’s conjecture for nonlocal minimal surfaces. arXiv:2306.07100 , 2023

arXiv 2023

[7] [7]

Caselli, E

M. Caselli, E. Florit-Simon, and J. Serra. Fractional S obolev spaces on R iemannian manifolds. Math. Ann. , 390(4):6249--6314, 2024

work page 2024

[8] [8]

Cabr\'e and Poggesi G

X. Cabr\'e and Poggesi G. Stable solutions to some elliptic problems: Minimal cones, the allen-cahn equation, and blow-up solutions. Lecture Notes in Mathematics: Geometry of PDEs and Related Problems , pages 1--45, 2018

work page 2018