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Higher order Weyl coefficients for the operator curl

This paper establishes refined spectral asymptotics for the curl operator on a closed Riemannian 3-manifold by deriving explicit formulae for the first six global and local Weyl coefficients, proving the holomorphy of the eta function in the half-plane $\operatorname{Re}s>-2$ , and improving Bär's counting function remainder to $o(\lambda^2)$ under specific geodesic flow assumptions.

Original authors: Giovanni Bracchi, Matteo Capoferri, Dmitri Vassiliev

Published 2026-07-07

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CC BY 4.0

Original authors: Giovanni Bracchi, Matteo Capoferri, Dmitri Vassiliev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are standing on a smooth, closed, three-dimensional surface (like a perfect sphere or a slightly squashed ball) that has no edges. On this surface, there is a mathematical "wind" or flow called curl. This isn't wind you can feel; it's a mathematical force that twists and turns.

The authors of this paper, Giovanni Bracchi, Matteo Capoferri, and Dmitri Vassiliev, are trying to understand the "notes" this wind can play. In math terms, these notes are called eigenvalues. Some notes are high-pitched (positive numbers), and some are low-pitched (negative numbers).

Here is the simple breakdown of what they did and what they found:

1. The Problem: Counting the Notes

Usually, when you look at a drum or a guitar string, the high notes and low notes are perfectly symmetrical. If you have a high note at 100, you might expect a low note at -100.

But for this specific "curl" wind, the notes are asymmetrical. The high notes and low notes don't mirror each other perfectly. The authors wanted to count how many notes exist below a certain pitch (a number called $\lambda$ ) as that pitch gets higher and higher.

They created two counters:

The Global Counter: Counts all the notes on the entire surface.
The Local Counter: Counts the notes in just one tiny spot on the surface.

2. The Main Discovery: The "Weyl Coefficients"

To count these notes accurately, mathematicians use a special recipe called an asymptotic expansion. Think of this like a recipe for a cake where you list the ingredients in order of importance:

Flour (The Big Ingredient): This is the main part of the count. It depends mostly on the size (volume) of the surface.
Sugar (The Next Ingredient): This depends on how curved the surface is.
Eggs, Butter, etc.: These are smaller, more subtle ingredients that tweak the recipe.

In this paper, the "ingredients" are called Weyl coefficients. The authors managed to write down the exact recipe for the first six ingredients.

Ingredients 1 & 2: They confirmed what was already known (the main count depends on volume).
Ingredients 3, 4, & 5: They calculated these exactly. Interestingly, the 3rd and 5th ingredients turned out to be zero (like a recipe that calls for no salt).
Ingredient 6: This is the big breakthrough. They found the sixth ingredient.

3. Why the Sixth Ingredient Matters

Here is the twist: The first five ingredients are the same whether you are counting the high notes or the low notes. The recipe looks identical for both sides.

However, the sixth ingredient is different for high notes versus low notes. It is the first time the "flavor" changes between the positive and negative sides. This proves mathematically why the notes are asymmetrical. It's like realizing that while your cake recipe is the same for chocolate and vanilla up to the fifth step, the sixth step uses a different spice that makes them taste distinct.

4. The "Eta Function" (The Symmetry Detector)

The paper also looks at something called the eta function. You can think of this as a "symmetry detector" that listens to all the notes at once.

Because the authors found that the first five "ingredients" cancel each other out perfectly, they proved that this symmetry detector works smoothly and doesn't break down until a very specific, deep level (mathematically, at a point called $s = -2$ ).
This confirms that the "glitch" in symmetry is very subtle and only appears deep in the math.

5. The "Geodesic Flow" Condition

To get the most precise count (refining the recipe to be even more accurate), the authors had to make a assumption about how "loops" work on the surface.

Imagine a ball rolling on your surface. If it rolls in a straight line and eventually comes back to exactly where it started, that's a "loop."
The authors assume that these loops are rare or "messy" enough that they don't clump together in a way that confuses the count. If the surface is "nice" enough (mathematically called "nonfocal" and "nonperiodic"), their recipe is perfect.

Summary

In plain English, this paper is a massive accounting job for a specific type of mathematical wind.

They built a super-precise calculator to count the "notes" of this wind.
They wrote down the first six terms of the formula used to count them.
They discovered that the sixth term is the first place where the "high notes" and "low notes" stop looking like mirror images of each other.
This allows them to prove that a specific mathematical "symmetry detector" (the eta function) works perfectly well for a wide range of values, only breaking down much later than previously thought.

They did this by treating the problem like a physics wave, tracking how the "wind" moves through time and space, and using advanced geometry to decode the pattern of the notes.

Technical Summary: Higher Order Weyl Coefficients for the Operator Curl

Problem Statement
This paper addresses the spectral geometry of the curl operator acting on real-valued 1-forms over a connected, oriented, closed Riemannian 3-manifold $(M, g)$ . Unlike the Laplace–Beltrami or Dirac operators, the curl operator is not elliptic and acts on vector fields (1-forms) rather than scalars or spinors, presenting significant technical challenges. The primary objective is to establish refined spectral asymptotics for the positive and negative eigenvalues of the curl operator separately. Specifically, the authors aim to compute explicit formulae for the first six global and local Weyl coefficients, which describe the asymptotic distribution of eigenvalues as the spectral parameter $\lambda \to +\infty$ . A secondary goal is to analyze the spectral asymmetry of the operator and determine the holomorphy domain of its associated eta function.

Methodology
The authors employ a constructive microlocal analysis approach centered on the explicit construction of the propagator for the curl operator. The core strategy involves the following steps:

Propagator Construction: The authors define the time-dependent propagator $U(t) = e^{-it \text{curl}}$ and decompose it into positive ( $U_+$ ), negative ( $U_-$ ), and zero-mode ( $U_0$ ) components. To handle the non-ellipticity of the curl operator, they introduce an auxiliary unitary operator $V(t) = U_+(t) + U_-(t) + U_0$ acting on the full Hilbert space of square-integrable 1-forms.
Oscillatory Integrals: The operator $V(t)$ is constructed as a sum of three oscillatory integrals (two time-dependent Fourier integral operators and one time-independent pseudodifferential operator) modulo infinitely smoothing operators ( $\Psi^{-\infty}$ ).
Transport Equations: The symbols of these oscillatory integrals are determined by solving a hierarchy of transport equations derived from the equation $(-i\partial_t + \text{curl})V = 0$ $(- i \partial_{t} + curl) V = 0$ . The algorithm involves:
- Identifying the principal symbol of curl and its eigenvalues ( $0, \pm \|\xi\|$ ).
- Solving for phase functions $\phi^{(\pm)}$ associated with the Hamiltonian flows of the non-zero eigenvalues.
- Solving algebraic and ordinary differential equations for the amplitude symbols $v^{(\pm)}_{-j}$ recursively.
- Addressing the overdetermined nature of the system for the zero-mode component by utilizing the properties of the extended curl operator (which is elliptic) to justify the construction.
Gauge Invariance: The authors carefully manage the gauge freedom inherent in the choice of eigencovectors for the principal symbol, ensuring that the final Weyl coefficients are gauge-invariant geometric invariants.
Asymptotic Extraction: By evaluating the mollified counting functions (convolutions of the spectral counting functions with a smooth test function), the authors extract the coefficients of the asymptotic expansion in powers of $\lambda$ .

Key Contributions and Results

Explicit Weyl Coefficients: The paper provides explicit formulae for the first six local Weyl coefficients, $c^{\pm}_k(y)$ $c_{k}^{\pm} (y)$ , for $k = 2, 1, 0, -1, -2, -3$ $k = 2, 1, 0, - 1, - 2, - 3$ .
- $c^{\pm}_2(y) = \frac{1}{2\pi^2}$ (Leading term, related to volume).
- $c^{\pm}_1(y) = 0$ .
- $c^{\pm}_0(y) = -\frac{1}{12\pi^2} \text{Sc}(y)$ (Proportional to scalar curvature).
- $c^{\pm}_{-1}(y) = 0$ .
- $c^{\pm}_{-2}(y) = -\frac{1}{80\pi^2} \|\mathring{\text{Ric}}(y)\|^2 + C (\Delta \text{Sc})(y)$ , where $\mathring{\text{Ric}}$ is the trace-free Ricci tensor and $C$ is a universal constant.
- $c^{\pm}_{-3}(y) = \mp \frac{1}{120\pi^2} E_{\alpha\beta\gamma}(y) \mathring{\text{Ric}}_{\alpha\mu}(y) \nabla_\beta \mathring{\text{Ric}}_{\gamma\mu}(y)$ .
Spectral Asymmetry: The authors demonstrate that spectral asymmetry (the difference between positive and negative spectral distributions) first manifests at the level of the sixth Weyl coefficient ( $k=-3$ ). The first five coefficients are symmetric or zero, meaning the leading terms of the asymptotic expansion do not distinguish between positive and negative spectra.
Refined Asymptotics: Under geometric conditions regarding the non-focality of points and the non-periodicity of geodesics, the remainder term in the counting function asymptotics is improved from $O(\lambda^2)$ to $o(\lambda^2)$ .
Eta Function Holomorphy: By analyzing the residues of the local eta function $\eta(y; s)$ , the authors prove that the eta function of the curl operator is holomorphic in the complex half-plane $\text{Re } s > -2$ . This refines previous conjectures suggesting holomorphy for $\text{Re } s > -1/2$ .
Symmetry Properties: A rigorous proof is provided for the symmetry relation $c^+_k(y) = (-1)^k c^-_k(y)$ , a property previously considered folklore but lacking a rigorous proof for the curl operator in the literature.

Significance
The paper claims to be the first to establish the sixth Weyl coefficients for the curl operator, which are the lowest-order coefficients exhibiting spectral asymmetry. This result is significant because it provides a quantitative measure of spectral asymmetry using high-energy eigenvalues, complementing existing studies that focus on low eigenvalues. The explicit formulae for the coefficients allow for the computation of the eta invariant and the residues of the eta function, which are crucial invariants in index theory and mathematical physics. Furthermore, the work bridges a gap in the spectral geometry of non-elliptic operators, offering a robust algorithmic framework that can be adapted to other first-order operators acting on forms. The results also improve the precision of Bär's earlier asymptotic formulae for the counting functions, refining the error term under appropriate geometric assumptions.