Spectral rigidity of Liouville tori

TL;DR

This paper proves that within a certain class of Liouville metrics on the torus, any Laplace isospectral deformation linear in the parameter must be trivial, using wave trace analysis and variational formulas.

Contribution

It establishes spectral rigidity for linear deformations of Liouville tori, linking length spectrum detection to triviality of isospectral deformations.

Findings

Laplace isospectral deformations are trivial under linearity assumption.

Wave trace noncancellation helps recover length spectrum information.

Preservation of a rational torus implies deformation triviality.

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.