Simultaneous metric uniformization of foliations by Riemann surfaces |
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Authors: | Email author" target="_blank">A A?GlutsyukEmail author |
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Institution: | (1) CNRS Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée dItalie, 69364 Cedex 07 Lyon, France |
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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
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
two-dimensional foliation on
compact manifold is wrong in general, even for suspensions over
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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Keywords: | 53C12 30F10 58F18 |
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