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Spectral rigidity of two-dimensional Liouville tori

This paper establishes two rigidity results for Laplace-isospectral deformations of generic Liouville metrics on the two-dimensional torus, proving that affine deformations are trivial and analytic deformations with trigonometric polynomial coefficients are merely componentwise rearrangements of the original metric.

Original authors: Joscha Henheik, Vadim Kaloshin, Yunzhe Li, Amir Vig

Published 2026-07-21

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CC BY 4.0

Original authors: Joscha Henheik, Vadim Kaloshin, Yunzhe Li, Amir Vig

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 holding a drum. If you strike it, it sings a specific set of musical notes, a unique "voice" determined entirely by its shape and the material it's made of. For decades, mathematicians have asked a fascinating question: If you hear the song, can you figure out exactly what the drum looks like? This is the heart of "spectral rigidity." It's like trying to guess the shape of a hidden object just by listening to the echoes it makes. Usually, if you change the shape of the drum even a tiny bit, the song changes. But what if you could twist and stretch the drum in a very specific way, keeping the song exactly the same? Does that mean the drum is secretly the same shape, just rearranged? This question becomes even trickier when the "drum" isn't a flat circle but a donut-shaped surface called a torus, and the way sound travels across it follows a special, orderly pattern known as "integrable flow."

This paper dives into that exact puzzle on a two-dimensional donut. The authors are investigating a special type of geometry called a "Liouville metric." You can think of these as a very specific, highly organized way of stretching the donut's surface, where the stretching in one direction depends only on that direction, and the stretching in the other depends only on the other. There is a long-standing guess in the math world that these Liouville metrics are the only way to make a donut where the paths of rolling balls (geodesics) stay perfectly orderly and predictable. The big question is: If you have a Liouville donut and you try to wiggle its shape while keeping its "song" (the Laplace spectrum) exactly the same, does it have to stay a Liouville donut? Or can you turn it into something completely different?

The authors prove that, for most Liouville donuts, the answer is a strict "no." You cannot secretly change the shape without changing the song, unless you are just shuffling the parts around in a very specific way. They show two main things. First, if you try to stretch the donut in a simple, straight-line fashion (a linear deformation) while keeping the song the same, you aren't actually changing anything at all; the stretch must be zero. It's like trying to push a wall that doesn't budge. Second, if you try a more complex, curved stretch (an analytic deformation), the only way to keep the song the same is if you are simply rearranging the "texture" of the donut. Imagine the donut is made of a fabric with a pattern of hills and valleys. You can slide those hills and valleys around, or swap the height of a hill in one spot with a hill in another, as long as you keep the total amount of "high ground" and "low ground" the same. This is called a "rearrangement."

However, the paper also warns that this isn't a magic trick where you can turn the donut into a cube and keep the song. They explicitly rule out the idea that you can create a completely new, non-Liouville shape that sounds the same. While they found a clever example of "two rivers" where you can slide strips of the surface around to keep the length of paths the same, they argue that this doesn't work for the full song (the Laplace spectrum). In fact, they believe those "river" examples would actually sound different if you listened closely enough. So, the conclusion is that for these special donuts, the song is a very strict fingerprint. If the song doesn't change, the underlying structure hasn't really changed either, except for a harmless reshuffling of its internal features. The authors are quite sure about this for the specific types of donuts they studied, proving it with rigorous math rather than just guessing or simulating. They haven't solved the problem for every possible shape in the universe, but for this specific, orderly class of donuts, they've shown that the song is a rigid lock that only opens if you don't change the key.

Technical Summary: Spectral Rigidity of Two-Dimensional Liouville Tori

Problem Statement
The paper addresses the spectral rigidity of Riemannian metrics on the two-dimensional torus ( $T^2$ ) whose geodesic flows are integrable. Specifically, it investigates the folklore conjecture that Liouville metrics are the unique integrable metrics on $T^2$ . The authors study whether a Liouville metric is determined by its Laplace spectrum (isospectrality) within a fixed conformal class. A Liouville metric is defined by a line element of the form:
$ds^2 = (1 + f_1(x_1) + f_2(x_2))(dx_1^2 + dx_2^2)$
The central question is: If a one-parameter family of metrics $(g_\epsilon)$ is isospectral (sharing the same Laplace eigenvalues) and starts as a Liouville metric, must the deformation be trivial or a specific type of rearrangement?

Methodology
The authors employ a combination of microlocal analysis, dynamical systems theory, and variational calculus. The proof strategy is divided into two main results, each relying on distinct technical ingredients:

Microlocal Trace Formula and Noncancellation:
- To link the Laplace spectrum to the length spectrum, the authors construct a microlocal parametrix for the wave propagator $\cos(t\sqrt{-\Delta})$ on the universal cover $\mathbb{R}^2$ .
- They utilize a WKB algorithm with Fourier integral operators to derive a Marvizi-Melrose style trace formula. This formula relates the singularities of the wave trace to the lengths of closed geodesics.
- A critical component is a "noncancellation" result. In integrable systems, wave trace singularities can theoretically cancel out if orbits with different Maslov indices conspire. The authors prove that under a "Noncoincidence Condition" (NCC)—which ensures lengths of orbits with conjugate points do not overlap with lengths of orbits without conjugate points (specifically "graph" rational tori)—the Laplace spectrum uniquely determines the lengths of these specific rational geodesics.
Variational Analysis:
- Linear Deformations: For the first main result, the authors assume the deformation is affine in $\epsilon$ ( $U(\epsilon, x) = \epsilon U(x)$ ). They analyze the second variation of the energy functional along closed geodesics. By showing that the preservation of a single rational torus and its length forces the first-order perturbation to vanish, they establish rigidity.
- Analytic Deformations: For the second main result, they consider analytic deformations where Taylor coefficients are trigonometric polynomials. They use the preservation of rational tori to show that the perturbation must be separable (i.e., $U(\epsilon, x_1, x_2) = U_1(\epsilon, x_1) + U_2(\epsilon, x_2)$ ).
- Rearrangement Analysis: Finally, they characterize the set of length-isospectral Liouville metrics. By mapping the length data to a real-analytic curve $\Gamma_{f_1, f_2}$ in $\mathbb{R}^2$ derived from the Maupertuis principle, they prove that preserving the marked length spectrum implies that the functions $f_1$ and $f_2$ are merely rearrangements of the original functions (up to an additive constant).

Key Contributions and Results

Theorem 1.1 (Rigidity of Affine Conformal Deformations):
For a generic Liouville metric $g_0$ , any isospectral deformation of the form $ds^2_\epsilon = (1 + f_1 + f_2 + \epsilon U)(dx_1^2 + dx_2^2)$ is trivial ( $U \equiv 0$ ). This result holds provided the deformation is linear in $\epsilon$ and the initial metric satisfies the Noncoincidence Condition.
Theorem 1.2 (Analytic Deformations and Rearrangements):
For a generic Liouville metric $g_0$ , any analytic isospectral deformation (with trigonometric polynomial coefficients) preserves the Liouville structure. Specifically, the perturbation $U$ is separable, and the resulting metric is obtained by a componentwise rearrangement of the original functions $f_1$ and $f_2$ .
- Note: The authors explicitly state that non-trivial length-isospectral deformations exist via rearrangements (e.g., the "Two Rivers" example), but they conjecture these are not Laplace isospectral.
Theoretical Tools:
- Theorem 2.1: Establishes that Laplace isospectrality implies the preservation of rational invariant tori and their lengths, provided the Noncoincidence Condition holds.
- Theorem 2.4: Proves that within the class of Liouville metrics, length isospectrality is equivalent to componentwise rearrangement of the conformal factors.

Significance and Claims
The paper claims to provide some of the first Laplace spectral rigidity results for closed manifolds with integrable geodesic flows.

Closed Manifold Analogue: The results serve as a closed manifold analogue to previous rigidity results for billiards and domains with boundaries (e.g., works by Hezari-Zelditch on ellipses).
Distinction from Dynamical Rigidity: Unlike dynamical rigidity results (which often require the preservation of the entire length spectrum or specific dynamical properties), this work derives rigidity from the Laplace spectrum alone, albeit under specific genericity and linearity assumptions.
Limitations and Scope: The authors are modest about the scope, noting that the results are currently restricted to dimension $d=2$ due to the reliance on transversality arguments for planar curves and specific trace formula theorems (Soga's theorem) that do not generalize to $d \ge 3$ . They also acknowledge that the linearity assumption in Theorem 1.1 might be necessary, citing the "Two Rivers" example where non-linear rearrangements preserve length spectra but likely not Laplace spectra.
Future Directions: The paper frames these results as a step toward a broader conjecture (Conjecture 1.3) that Liouville metrics are spectrally rigid in a neighborhood (i.e., any metric sufficiently close and isospectral is isometric). They also raise the question of whether integrable metrics near Liouville metrics must themselves be Liouville (Question 1.4).

In summary, the paper demonstrates that for generic Liouville metrics on the 2-torus, the Laplace spectrum is a strong invariant that restricts deformations to trivial ones (in the linear case) or rearrangements (in the analytic case), effectively supporting the folklore conjecture that Liouville metrics are the unique integrable metrics up to these specific transformations.

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