Deformational rigidity of integrable metrics on the torus
Joscha Henheik
TL;DR
The paper studies deformational rigidity of Liouville metrics on ${\mathbb T}^2$ under conformal integrable perturbations. By translating the deformational problem via the Maupertuis principle into perturbations of a mechanical Hamiltonian and leveraging action-angle coordinates, a harmonic-first-order equation for the perturbation is obtained. Under preservation of a rich set of rational invariant tori, the authors show that the perturbing potential must be separable, i.e. $U(x)=U_1(x^1)+U_2(x^2)$, implying that integrable deformations in the same conformal class remain Liouville metrics; this is extended to higher dimensions and to analytic perturbations with exceptional-null-measure sets. The results tie deformational rigidity to Fourier-analysis-based full-rank conditions and connect with broader rigidity phenomena in integrable systems and perturbative Birkhoff-type problems, offering a framework for understanding when integrable perturbations preserve Liouville structure.
Abstract
It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: We consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations the deformed metric is again Liouville. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. In order to put our results in perspective, we review existing results about integrable metrics on the torus.