Article Dans Une Revue Commentarii Mathematici Helvetici Année : 2004

Simultaneous metric uniformization of foliations by Riemann surfaces

Alexey Glutsyuk

Résumé

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 T2×R and families of complex structures with bounded derivatives C3- close to the standard complex structure. We prove that the analogous statement for arbitrary Cinfty two-dimensional foliation on compact manifold is wrong in general, even for suspensions over T2: 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.
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Dates et versions

ensl-00078827 , version 1 (07-06-2006)

Identifiants

  • HAL Id : ensl-00078827 , version 1

Citer

Alexey Glutsyuk. Simultaneous metric uniformization of foliations by Riemann surfaces. Commentarii Mathematici Helvetici, 2004, 79 (4), pp.704-752. ⟨ensl-00078827⟩
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