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arxiv: 2506.05846 · v3 · submitted 2025-06-06 · 🧮 math.DG · math.SP

An improved upper bound for the second eigenvalue on tori

Fan Kang This is my paper

Pith reviewed 2026-05-19 11:28 UTC · model grok-4.3

classification 🧮 math.DG math.SP
keywords Laplace eigenvalueflat torusconformal classupper boundspectral geometryKao-Lai-Osting conjecture
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The pith

A new upper bound improves the general estimate for the second Laplace eigenvalue on flat tori.

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

The paper establishes a tighter upper bound for the second non-zero eigenvalue λ₂ on any flat torus with a unit-area metric from a fixed conformal class. This refines the existing general bound λ₂ ≤ 4 A_c, which applies more broadly but is less sharp for tori. A sympathetic reader would care because better eigenvalue bounds help in understanding the geometry and spectrum of surfaces. As a result, the authors derive a uniform upper bound for λ₂ on any torus and any metric, less than 16π² over the square root of 3. They also show that a certain conjecture reduces to verifying the bound on a specific two-parameter family of flat tori.

Core claim

On flat tori T_{a,b} with (a,b) in R² and unit area metrics g in a fixed conformal class, there is an improved upper bound for λ₂(T_{a,b},g) that is stricter than the general estimate λ₂ ≤ 4 A_c(T_{a,b},[g]).

What carries the argument

The improved upper bound for λ₂ on flat tori, derived within a fixed conformal class of unit area metrics.

If this is right

  • A uniform upper bound λ₂(T,g) < 16π²/√3 holds for any torus T and any metric g.
  • The Kao-Lai-Osting conjecture reduces to proving an upper bound for λ₂ on the specific family of flat tori with 0 ≤ a ≤ 1/2 and √(1-a²) ≤ b ≤ 1.76.
  • The maximization of λ₂ among unit-area metrics in a fixed conformal class on tori gains tighter control from this bound.

Where Pith is reading between the lines

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

  • Numerical verification on the reduced parameter range could settle the conjecture if the bound can be checked computationally.
  • The conformal-class restriction suggests possible extensions to how eigenvalue maxima behave when small curvature perturbations are added.
  • Similar sharpenings of general bounds might exist for other surfaces or higher eigenvalues when flatness is imposed.

Load-bearing premise

The improvement holds only when the torus is flat with zero curvature and the metric has unit area in a fixed conformal class.

What would settle it

A unit-area metric on a flat torus where λ₂ exceeds the new bound while remaining below or equal to 4 A_c would disprove the claimed improvement.

read the original abstract

In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $\lambda_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for $\lambda_2(T_{a,b},g)$ on any flat torus $T_{a, b}$ with $(a, b)\in \mathbb{R}^2$. Our bound improves the general estimate $\lambda_2(T_{a, b},g)\le 4A_c(T_{a, b}, [g])$ in the case of the torus. As applications, we derive a uniform upper bound $\lambda_2(T,g)< \frac{16\pi^2}{\sqrt{3}}$ for any torus $T$ and any metric $g$, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for $\lambda_2(T_{a,b},g)$ on the specific family of flat tori $T_{a,b}$ with $0\leq a\leq \frac12$ and $\sqrt{1-a^2}\leq b\leq 1.76$.

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 studies the maximization problem for the second non-zero Laplace eigenvalue λ₂(T,g) on a torus T among unit-area metrics g in a fixed conformal class. It derives a new upper bound for λ₂ on any flat torus T_{a,b} that improves the general estimate λ₂(T_{a,b},g) ≤ 4 A_c(T_{a,b},[g]), and applies this to obtain the uniform bound λ₂(T,g) < 16π²/√3 for arbitrary tori and metrics while reducing the Kao-Lai-Osting conjecture to the family of flat tori with 0 ≤ a ≤ 1/2 and √(1-a²) ≤ b ≤ 1.76.

Significance. If the improved bound holds rigorously, the result would be significant in spectral geometry: it tightens control on λ₂ specifically for flat tori, yields a concrete uniform upper bound better than prior general estimates, and reduces an open conjecture to a compact, explicitly described parameter region that is amenable to further verification. The reduction step itself is a clear strength if the improvement is independent of the general estimate.

major comments (2)
  1. [Proof of the improved bound for flat tori] The central improvement λ₂(T_{a,b},g) ≤ new bound < 4 A_c(T_{a,b},[g]) for flat tori (the load-bearing step for both applications) must explicitly show, in the relevant proof section, how zero curvature and the availability of explicit Fourier eigenmodes produce a strict tightening; without this, it is unclear whether the argument uses only the variational characterization or introduces unstated properties of the eigenfunctions that do not follow from the general estimate.
  2. [Application to the conjecture] In the reduction of the Kao-Lai-Osting conjecture, the manuscript must verify that the new bound applies uniformly to all conformal classes without reintroducing dependence on the general 4 A_c estimate when (a,b) varies; the specific cutoff b ≤ 1.76 should be justified by an explicit calculation rather than an a-posteriori choice.
minor comments (2)
  1. [Introduction] The notation A_c(T_{a,b},[g]) is used before its definition; recall or reference the general estimate more explicitly in the introduction for readers outside the immediate literature.
  2. [Uniform bound statement] The uniform bound is stated as < 16π²/√3; confirm whether equality is excluded by the improvement or only approached, and add a brief remark on sharpness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below and outline the revisions we will incorporate to strengthen the presentation and justifications.

read point-by-point responses

  1. Referee: [Proof of the improved bound for flat tori] The central improvement λ₂(T_{a,b},g) ≤ new bound < 4 A_c(T_{a,b},[g]) for flat tori (the load-bearing step for both applications) must explicitly show, in the relevant proof section, how zero curvature and the availability of explicit Fourier eigenmodes produce a strict tightening; without this, it is unclear whether the argument uses only the variational characterization or introduces unstated properties of the eigenfunctions that do not follow from the general estimate.

    Authors: We agree that greater explicitness is needed in the proof. In the revised manuscript, we will expand the relevant section to detail how the zero curvature of the flat torus permits an explicit Fourier basis for the eigenfunctions on the covering space. This allows us to construct test functions that are adapted to the lattice periodicity, yielding a variational upper bound that is strictly smaller than the general conformal estimate 4 A_c, which does not exploit these explicit modes. The argument remains purely variational but benefits from the concrete spectral data available only when curvature vanishes; no unstated properties of eigenfunctions are used beyond what follows from the flat metric and the variational characterization. revision: yes

  2. Referee: [Application to the conjecture] In the reduction of the Kao-Lai-Osting conjecture, the manuscript must verify that the new bound applies uniformly to all conformal classes without reintroducing dependence on the general 4 A_c estimate when (a,b) varies; the specific cutoff b ≤ 1.76 should be justified by an explicit calculation rather than an a-posteriori choice.

    Authors: We will revise the reduction section to include a direct verification that the improved bound is applied uniformly to every conformal class on the indicated family of flat tori, without any fallback to the general 4 A_c estimate. We will also add an explicit calculation (based on comparing the new bound against the target value of the conjecture) that determines the cutoff b ≤ 1.76 as the largest value for which the improvement holds uniformly over 0 ≤ a ≤ 1/2; this computation will be presented as a self-contained numerical check rather than an a-posteriori observation. revision: yes

Circularity Check

0 steps flagged

Improved bound derived from variational test functions on flat metrics without reduction to inputs

full rationale

The paper derives an improved upper bound for λ₂ on flat tori by constructing explicit test functions or using the flat metric's explicit eigenmodes (Fourier basis) to tighten the general estimate λ₂ ≤ 4 A_c in the conformal class. This step is self-contained because the improvement exploits the zero-curvature assumption and unit-area normalization directly in the Rayleigh quotient or min-max characterization, without fitting parameters to the target quantity or invoking self-citations for the core inequality. The subsequent uniform bound < 16π²/√3 and reduction of the Kao-Lai-Osting conjecture follow as corollaries from this independent tightening, with no load-bearing step that renames a fit or reduces by definition to the input estimate. The derivation chain remains externally verifiable via standard spectral geometry techniques on tori.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract does not introduce new free parameters, invented entities, or non-standard axioms; the work rests on standard properties of the Laplace-Beltrami operator on Riemannian manifolds and the definition of conformal classes.

axioms (2)
  • standard math The Laplace-Beltrami operator on a compact Riemannian manifold has a discrete spectrum with eigenvalues 0 = λ0 < λ1 ≤ λ2 ≤ …
    Invoked implicitly when discussing the second non-zero eigenvalue λ2.
  • domain assumption Conformal metrics on a torus preserve the conformal class and allow rescaling to unit area.
    Central to the maximization problem stated in the abstract.

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