Smoothness of the uniformization of two-dimensional linear foliation on torus with nonstandard metric

Abstract

Consider a parallel plane foliation on real finite-dimensional linear vector space. It induces a foliation on the torus obtained by factorization of the space by the integer lattice (let us denote the latter foliation by F). Let g be arbitrary metric on the torus. It induces a complex structure on each leaf of F such that all the leaves are parabolic: each individual leaf admits a conformal flat complete metric. We show that for any foliation F as above and arbitrary smooth metric g the correspondent complete conformal flat metric on the leaves, which is smooth on individual leaf, can be chosen to be smooth in the whole torus (in particular, in transversal parameter). We show that under appropriate Diophantine condition on the foliation for arbitrary smooth metric g there exists a smooth Euclidean metric on the torus conformal on the leaves such that the latters are totally geodesic; in other terms, the correspondent triple consisting of the torus, the foliation and the complex structure on the leaves is isomorphic to a triple correspondent to a linear foliation on another torus equipped with the standard complex structure on the leaves. This Diophantine condition is exact. We state and prove the analytic versions of these statements.