Zoll magnetic systems on the two-torus: A Nash–Moser construction

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin [19] on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional S, and then look for its zeros by means of a Nash–Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator dS in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator dS∘dS⁎ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of dS∘dS⁎ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of dS∘dS⁎ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.