Deformational rigidity of integrable metrics on the torus

TL;DR

This paper investigates the stability of integrable Liouville metrics on the torus under conformal deformations, showing that many such deformations preserve integrability and extending results to higher dimensions.

Contribution

It demonstrates that a large class of conformal deformations of Liouville metrics on the torus remain Liouville, generalizing the result to higher-dimensional tori.

Findings

Deformed metrics are often still Liouville.

Method extends to higher-dimensional tori.

Supports conjecture on rigidity of integrable metrics.

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.