Continuity of the uniformization function of linear foliation on three-dimensional torus with nonstandard metric
by
Alexei A. Glutsuk
Independent University of Moscow

Denote T³=R³\slash2\piZ³. Consider a parallel plane foliation on R³. The standard projection R³ --> T³ induces a foliation on the torus T³. This foliation will be denoted by F.

Let g be arbitrary (smooth) Riemann metric on T³. Its restriction to the leaves of the foliation F defines a complex structure on them (it will be denoted by \sigma_g). The leaves are parabolic Riemann surfaces with respect to \sigma_g: their universal coverings are conformally equivalent to complex plane. This follows from Riemann quasiconformal mapping theorem and boundedness of the dilatation of the metric g with respect to the standard Euclidean metric of T³. Thus, each leaf admits a complete \sigma_g- conformal flat metric. This means that there exists a real positive function \phi on the leaf such that the metric \phig on the leaf is flat and complete. This function is unique up to multiplication by constant.

E.Ghys stated the following problem: is it true that for any smooth metric g on the torus T³ the correspondent function \phi (which defines the flat metric \phig on the leaves) may be chosen to be continuous (smooth) on the whole torus: continuous (smooth) not only in the parameter of an individual leaf, but also in the transversal parameter (in particular, in all the intersections of the given leaf with transversal circle)?

In the simplest case, when the leaves of the foliation F are tori, the answer to this question is positive. This follows from the classical theorem on dependence of uniformization on parameter of complex structure .

We prove the following Theorem giving the positive answer to Ghys's question.

Theorem 1 Let F be as at the beginning of the abstract, g be an analytic /C^\infty /finite dilatation measurable/ metric on T³. There exists an analytic /C^\infty /measurable/ positive function \phi:T³ --> R₊ such that the restriction of the metric \phig to the leaves of the foliation F is flat.

Theorem 1 is implied by the following more general statement.

Lemma 1 In the conditions of Theorem 1 let \sigma_g be the complex structure on the leaves such that the restriction to them of the metric g is conformal with respect to \sigma_g. There exists a nowhere vanishing analytic /C^\infty /L₂/ differential 1- form \omega_g on T³ such that its restriction to each leaf is \sigma_g- holomorphic and moreover there exists a quasiconformal homeomorphism of the leaf onto complex plane with the derivative \omega_g transforming \sigma_g to the standard complex structure.

For a foliation F satisfying a Diophantine condition we prove an analogue of Riemann mapping theorem. To formulate it, let us introduce the two following Definitions.

We say that a number \alpha in R\Q is Diophantine, if there exist constants C, \epsilon > 0 such that for any pair p, q in Z, q =/= 0, the following inequality holds:

|\alpha-\frac pq| > \frac Cq2+\epsilon.

Let x=(x₁, x₂, x₃) be coordinates in the space R³. Consider the foliation on R³ by level planes of the linear function l(x)=\alpha₁x₁+\alpha₂x₂-x₃. Let F be the correspondent factorized foliation on T³=R³\slash 2\piZ³. We say that F is Diophantine, if the additive subgroup in R generated by \alpha₁ and \alpha₂ contains a Diophantine number.

Lemma 2 Let F be a Diophantine foliation (see Definition 2), g be an analytic (C^\infty) metric on T³, \sigma_g be the complex structure on the leaves of F such that g is \sigma_g- conformal. There exists a discrete rank 3 additive subgroup G subset R³ and an analytic (C^\infty) diffeomorphism T³ --> T_G=R³\slash G that transforms F to a linear foliation and \sigma_g to the standard complex structure induced by the Euclidean metric.

One can provide examples of foliations F as at the beginning of the abstract with dense leaves such that for a generic analytic metric the statement of Lemma 2 is false.

Earlier a particular case of the problem on existence of continuous complete conformal flat metric stated at the beginning of the abstract was studied by E.Ghys . He proved the statement of Theorem 1 under the additional assumption that in the notations of Definition 2 the additive group generated by the numbers \alpha₁ and \alpha₂ contains either a nonzero rational, or a Diophantine number.

References 1. Ghys, E. Sur l'uniformisation des laminations paraboliques. - in Integrable systems and foliations, ed. C.Albert, R.Brouzet, J.-P. Dufour (Montpellier, 1995), Progress in Math. 145 (1996), 73-91.

2. Abikhow, W. Real analytic theory of Teichmüller space, - Lect. Notes in Math., 820, Springer-Verlag (1980).

3. Ahlfors, L. Lectures on quasiconformal mappings, Wadsworth (1987).

Date received: August 11, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-51.