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Topology Vol.16, pp. 13-32. Pergamon Press, 1977. Printed in Great Britain FOLIATIONS WITH ALL LEAVES COMPACT*
 

Summary: Topology Vol.16, pp. 13-32. Pergamon Press, 1977. Printed in Great Britain
FOLIATIONS WITH ALL LEAVES COMPACT*
ROBERTEDWARDS,KEr~Ern MILLETrand DENNISSULLIVAN
(Received20 October1975)
1.
THE PLr~OSEof this paper is to present some information about the following Question:
I[M is a compact manifold [oliated by compact submani[olds (everything smooth), is there an
upper bound on the volume of the leaves?t
In particular, if M is a compact manifold supporting a nonsingular flow in which each orbit
is periodic, is there an upper bound on the lengths of the orbits? In his thesis, Reeb ([12], for an
analytic example see [3, p. 68]) describes a smooth flow on a non-compact manifold such that all
orbits are periodic and such that the lengths of the orbits are not locally bounded. After our
research was completed, the third author found a smooth flow on a closed 5-manifold[15, 16],
which showed that the answer, in.general, was no. The former example shows that the question
is global and cannot be answered by simply considering the structure of the foliation in a
neighborhood of individual compact leaves. The latter example shows that some additional
hypothesis on M is required. This example is worth keeping in mind while reading this paper due
to its close connection with our main result.
The existence of an upper bound on the volume of the leaves has rather important
consequences which provide a description of the local, as well as global, structure of the

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics