Extension of Levi-flat hypersurfaces past CR boundaries

Details Extension of Levi-flat hypersurfaces past CR boundaries Extension of Levi-flat hypersurfaces past CR boundaries

2006-12-03

arxiv.org/abs/math/0612071v4

Indiana Univ. Math. J., 57 (2008), no. 2, 699-716

Mathematics

Complex Variables

15 pages; latex, amsrefs; fix statement and proof of theorem 3.1; accepted in Indiana Univ. Math. J

Scientific paper

10.1512/iumj.2008.57.3203

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the hypersurface to be real analytic. This allows us to classify all real analytic generic boundaries of Levi-flat hypersurfaces in terms of their normal coordinates. For the remaining case of generic real analytic boundary we get a weaker extension theorem. We find examples to show that these two extension results are optimal. Further, a class of nowhere minimal real analytic submanifolds is found, which is never the boundary of even a $C^2$ Levi-flat hypersurface.

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