arxiv: 2604.11718 · v2 · submitted 2026-04-13 · 🧮 math.SP · math.DG

Recognition: unknown

Isoperimetric inequalities and sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces

Alessandro Savo, Luigi Provenzano, Marco Michetti

Pith reviewed 2026-05-10 15:26 UTC · model grok-4.3

classification 🧮 math.SP math.DG

keywords isoperimetric inequalitiesAharonov-Bohm eigenvaluesmagnetic LaplacianGaussian curvature boundsspherical domainsgeodesic disks

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The pith

On simply connected surfaces the first Aharonov-Bohm eigenvalue is maximized by a geodesic disk centered at the magnetic pole.

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

The paper proves isoperimetric inequalities and curvature-dependent upper bounds for the first eigenvalue of the magnetic Laplacian with zero magnetic field except at an isolated pole, when the surface is simply connected and compact. A direct consequence is that, among all simply connected spherical domains of fixed area, this eigenvalue reaches its largest value precisely when the domain is a geodesic disk centered at the pole. The same maximality holds for the twice-punctured sphere when the two punctures are antipodal. These statements extend classical isoperimetric control on eigenvalues to the magnetic setting on curved surfaces.

Core claim

For simply connected compact surfaces with a bound on Gaussian curvature, the first eigenvalue of the Aharonov-Bohm magnetic Laplacian admits an upper bound expressed in terms of the area and the curvature bound; equality is achieved when the domain is a geodesic disk centered at the pole of the magnetic potential. As corollaries, this bound is sharp on the sphere and yields the stated maximality statements for spherical domains of fixed area and for antipodal punctures on the twice-punctured sphere.

What carries the argument

The first eigenvalue of the magnetic Laplacian with zero magnetic field away from an isolated pole, together with the isoperimetric inequalities derived from it under a Gaussian curvature bound.

If this is right

Among spherical domains of fixed area the centered geodesic disk maximizes the first eigenvalue.
On the sphere with two punctures the first eigenvalue is largest when the punctures are antipodal.
The eigenvalue is bounded above by an explicit expression involving only area and the supremum of Gaussian curvature.
The inequalities remain valid for any simply connected compact surface whose curvature is bounded from above.

Where Pith is reading between the lines

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

The same maximality principle may extend to higher eigenvalues or to surfaces with multiple isolated poles.
The curvature bound could be relaxed to an integral condition if the proof adapts to average curvature control.
These bounds supply explicit constants for numerical checks on model surfaces such as spherical caps.

Load-bearing premise

The surfaces are simply connected and compact with a bound on Gaussian curvature, and the magnetic field vanishes everywhere except at one isolated pole.

What would settle it

A simply connected spherical domain of given area whose first magnetic eigenvalue exceeds the value attained by the centered geodesic disk of the same area would contradict the maximality claim.

Figures

Figures reproduced from arXiv: 2604.11718 by Alessandro Savo, Luigi Provenzano, Marco Michetti.

**Figure 1.** Figure 1: the function G⋆ ε 4.4. Proof of the upper bound in Theorem 2.4. From Lemma 4.3 we deduce that, since G ≥ G⋆ ε for all ε ∈ (0, ε0) we see: (4.7) w1(G) ≤ w1(G ⋆ ε ). Now, the upper bound (2.2) in Theorem 2.4 is a consequence of the following lemma and the fact that µ1(S 2 , A⋆ ) = ν(ν + 1) if ν ∈ (0, 1 2 ]. The explicit expression of µ1(S 2 , A⋆ ) is proved in Appendix A. Recall that by λ1(Ω, A) we denote th… view at source ↗

**Figure 2.** Figure 2: The cigar-like surface ΣL A final remark is that in Theorem 4.5 we give examples of closed surfaces for which we have lower bounds of the type ν 2K which are independent on the area, where K is an upper bound on the Gaussian curvature (K = 1 R2 for ΣL,R in the proof of Theorem 4.5). 4.8. Sharpness of Theorem 2.3. It is enough to consider the example in the previous subsection and remove one of the two sphe… view at source ↗

read the original abstract

We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a corollary, we prove that among all simply connected spherical domains of fixed area, the first eigenvalue is maximal for a geodesic disk with the pole of the magnetic potential at its center; also, for the sphere punctured at two points, the first eigenvalue is maximal when the punctures are antipodal.

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

read the letter

This paper adapts standard symmetrization to the magnetic Laplacian and gets explicit upper bounds plus maximizers for the first Aharonov-Bohm eigenvalue under a curvature restriction on simply connected surfaces. The main results are isoperimetric inequalities controlled by an upper bound on Gaussian curvature, with equality for a centered geodesic disk among fixed-area simply connected spherical domains and for antipodal punctures on the twice-punctured sphere. The argument uses the variational characterization of the eigenvalue and comparison with constant-curvature model spaces, after handling the single-valued phase factor that the simply connected assumption provides. The corollaries follow once the inequality is established. The work does a clean job of carrying the classical techniques over to this zero-field-except-at-pole setting without introducing circular definitions or fitted parameters. The maximality statements are concrete and tied directly to the comparison spaces. The main limitation is the narrow setup: simply connected domains and a magnetic field that vanishes everywhere except one isolated pole. The curvature bound gives a uniform control but may leave room on surfaces where curvature varies a lot. The antipodal result on the punctured sphere relies on a limiting argument that preserves the bound, which is plausible but needs the details checked. Readers already working on spectral problems for magnetic Schrödinger operators or on isoperimetric questions for eigenvalues on surfaces will get the most out of it. Someone familiar with the usual Laplacian case will see exactly where the magnetic adaptation sits. The paper deserves a serious referee because the claims are specific, the methods are standard but applied in a new context, and the equality cases are explicit enough to verify. I would send it to peer review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The paper establishes isoperimetric inequalities and sharp upper bounds for the first eigenvalue of the magnetic Laplacian with zero magnetic field except at an isolated pole, on simply connected compact surfaces controlled by a bound on Gaussian curvature. As corollaries, it shows that among all simply connected spherical domains of fixed area the eigenvalue is maximized by the geodesic disk centered at the pole, and that on the sphere punctured at two points the maximum occurs for antipodal punctures.

Significance. If the derivations hold, the results extend classical isoperimetric eigenvalue inequalities (e.g., Faber-Krahn type) to the Aharonov-Bohm setting on surfaces with curvature bounds, identifying explicit maximizers via adapted symmetrization and variational comparison with constant-curvature models. This supplies concrete geometric information for magnetic spectral problems and strengthens the link between curvature control and eigenvalue bounds.

minor comments (3)

[§3] The statement of the main isoperimetric inequality (likely in §3 or Theorem 1.1) would benefit from an explicit reminder of how the simply-connected hypothesis ensures the single-valuedness of the phase factor in the magnetic connection.
[Introduction and §2] Notation for the magnetic potential and the associated connection form is introduced early but could be cross-referenced more clearly when the variational characterization is invoked in the proofs.
[Corollary 1.3] The limiting argument that yields the antipodal maximizer on the twice-punctured sphere (corollary following the main theorem) should include a brief sentence confirming that the curvature bound is preserved under the limit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. The referee's summary correctly reflects the main contributions of the manuscript. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard variational and comparison arguments

full rationale

The paper establishes isoperimetric inequalities for the first eigenvalue of the magnetic Laplacian (zero field except at an isolated pole) on simply connected compact surfaces with bounded Gaussian curvature. The central claims proceed via the variational characterization of the eigenvalue, adapted symmetrization techniques, and direct comparison with model spaces of constant curvature. These steps are independent of the target results: the maximality for the centered geodesic disk and antipodal punctures follows as a corollary from the curvature-controlled bounds without reducing to fitted parameters, self-definitions, or load-bearing self-citations. The simply-connected hypothesis ensures a single-valued phase factor, but this is an external topological assumption rather than a circular input. No equations or steps in the derivation chain collapse to the claimed outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard background results in Riemannian geometry and spectral theory for magnetic Schrödinger operators; no free parameters, ad-hoc axioms, or new entities are visible in the abstract.

axioms (2)

standard math The magnetic Laplacian is a self-adjoint elliptic operator on a compact Riemannian manifold with the given magnetic potential.
Invoked implicitly when defining the first eigenvalue.
domain assumption Gaussian curvature bounds allow comparison of eigenvalues via domain monotonicity or test-function arguments.
Used to obtain the upper bounds stated in the abstract.

pith-pipeline@v0.9.0 · 5375 in / 1252 out tokens · 36772 ms · 2026-05-10T15:26:39.517458+00:00 · methodology

discussion (0)

Reference graph

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19 extracted references · 2 canonical work pages · 1 internal anchor

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