• DOI:10.1090/S0002-9947-2010-04887-1
• Corpus ID: 14268868

Levi-flat hypersurfaces with real analytic boundary

@article{Lebl2007LeviflatHW,
  title={Levi-flat hypersurfaces with real analytic boundary},
  author={Jiř{\'i} Lebl},
  journal={Transactions of the American Mathematical Society},
  year={2007},
  volume={362},
  pages={6367-6380}
}

• Jiří Lebl
• Published 19 October 2007
• Mathematics
• Transactions of the American Mathematical Society

Let X be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold M of X, that is the boundary of a compact Levi-flat hypersurface H, we study the regularity of H. Suppose that the CR singularities of M are an O(X)-convex set. For example, suppose M has only finitely many CR singularities, which is a generic condition. Then H must in fact be a real analytic submanifold. If M is real algebraic, it follows that H is real algebraic and in fact extends past M, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic. We prove a similar theorem for submanifolds with boundary, and another one for subanalytic sets.