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Let $M$ be a compact 3-manifold with boundary equipped with a 1-dimensional foliation all of whose leaves are circles. An old theorem of Epstein says that $M$ is a Seifert fibered space.

The proof of Epstein’s theorem is quite complicated, and it predates the modern theory of 3-manifolds centered around Thurston’s work and geometrization.

Question: Are there any alternate proofs now of Epstein’s theorem?

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    $\begingroup$ I'm not sure that newer developments, such as the geometrization theorem, might help with this problem. The issue seems to be the boundedness of the lengths of the leaves, which fails for foliations by circles of a compact 5-manifold by a result of Sullivan. On the other hand, Edwards-Millet-Sullivan showed more generally that codimension-2 foliations of manifolds with compact leaves have leaves of bounded volume. doi.org/10.1016/0040-9383(77)90028-3 Note that they incorporate a simplification to part of Epstein's proof due to Norris Weaver. jstor.org/stable/1970855 $\endgroup$
    – Ian Agol
    Commented Jul 29, 2020 at 5:52

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