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This article is a part of a project investigating the relationship betweenthe dynamics of completely integrable or close to completely integrablebilliard tables, the integral geometry on them, and the spectrum of thecorresponding Laplace-Beltrami operators. It is concerned with new isospectralinvariants and with the spectral rigidity problem for families ofLaplace-Beltrami operators with Dirichlet, Neumann or Robin boundaryconditions, associated with C^1 families of billiard tables. We introduce anotion of weak isospectrality for such deformations. The main dynamicalassumption on the initial billiard table is that the corresponding billiardball map or an iterate of it has a Kronecker invariant torus with a Diophantinefrequency and that the corresponding Birkhoff Normal Form is nondegenerate inKolmogorov sense. Then we obtain C^1 families of Kronecker tori withDiophantine frequencies. If the family of the Laplace-Beltrami operatorssatisfies the weak isospectral condition, we prove that the average action onthe tori and the Birkhoff Normal Form of the billiard ball maps remain the samealong the perturbation. As an application we obtain infinitesimal spectralrigidity for Liouville billiard tables in dimensions two and three.Applications are obtained also for strictly convex billiard tables of dimensiontwo as well as in the case when the initial billiard table admits an ellipticperiodic billiard trajectory. Spectral rigidity of billard tables closeelliptical billiard tables is obtained. The results are based on a constructionof C^1 families of quasi-modes associated with the Kronecker tori and onsuitable KAM theorems for C^1 families of Hamiltonians.

From K.A.M. Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables