Simultaneous metric uniformization of foliations by Riemann surfaces

Abstract : We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on $\mathbb T^2\times\mathbb R$ and families of complex structures with bounded derivatives $C^3$- close to the standard complex structure. We prove that the analogous statement for arbitrary $C^ infty$ two-dimensional foliation on compact manifold is wrong in general, even for suspensions over $\mathbb T^2:$ in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded.