Simultaneous metric uniformization of foliations by Riemann surfaces (Q1884665)

Two-dimensional linear foliations of a torus \(\mathbb T^n\) are considered. The author proves that for any smooth family of complex structures on the leaves there exists a smooth family of uniformizing (conformal complete flat) metrics on the leaves. This result is extended to linear foliations of \(\mathbb T^2\times\mathbb R\) and to families of complex structures with bounded derivatives \(C^3\)-close to the standard complex structure. The author also proves that the analogous statement for arbitrary \(C^{\infty}\) two-dimensional foliations of a compact manifold is wrong in general.