Spectral rigidity of Liouville tori

TL;DR

The paper proves spectral rigidity for Liouville metrics on the 2-torus: any Laplace isospectral deformation within the conformal class ds^2_ε = (1 + f1(x1) + f2(x2) + εU(x1,x2))(dx1^2 + dx2^2) with linear ε-perturbations must have U ≡ 0. The authors combine a microlocal trace analysis, which shows isospectrality preserves rational invariant tori (hence the rotational length spectrum) with a second-variation argument that, given a preserved rational torus, forces the perturbation U to vanish along the corresponding geodesics, and then everywhere by foliation. The key innovations include a Hadamard-Riesz type parametrix for off-diagonal wave propagation, a Marvizi-Melrose style trace formula localized to horizontal tori, and the use of loop functions to connect the length spectrum and geodesic dynamics. Together these yield a two-step rigidity result: isospectrality implies rational integrability, and preserving a single rational torus under a linear perturbation implies trivial deformation, establishing spectral rigidity in this integrable closed-manifold setting with potential implications for broader inverse spectral problems in integrable dynamics.

Abstract

We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof are a noncancellation result for the wave trace and an analysis of the second order variational formula for the energy functional associated to closed geodesics. Noncancellation allows us to detect parts of the length spectrum from the Laplace spectrum and conclude rational integrability for the deformed geodesic flow (Liouville metrics are folklorically conjectured to be the only Riemannian metrics with integrable geodesic flow on the torus). We then use the second variational formula to show how the preservation of a single rational torus is sufficient to conclude triviality of the deformation, assuming linearity. We also present some evidence that our hypothesis of linearity may indeed be necessary.

Figures (4)

• Figure 1: Two rivers (Example \ref{['ex:two rivers']}): Perturbing the flat metric along horizontal blue strips (the rivers), we can then translate one of these rivers while preserving the length spectrum of the metric. More precisely, by varying the distance between the two blue regions, we produce nonisometric perturbations which are nonetheless length isospectral.
• Figure 2: An approximation of the wave trace on $\mathbb{R}^2/\mathbb{Z}^2$ with the Euclidean metric, made with finitely many eigenvalues.
• Figure 3: Proving the Lipschitz property of rational tori
• Figure 4: Proving the twist property of rational tori

Theorems & Definitions (52)

• Theorem 1.1: Main result
• Conjecture 1.2
• Example 1.4: Two rivers
• Theorem 1.5: Isospectrality and rational integrability
• Theorem 1.6: Rigidity by preservation of one rational invariant torus
• proof : Proof of Theorem \ref{['thm:main2']}
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Remark 3.4