Levi-flat hypersurfaces with real analytic boundary

Lebl, Jiří

doi:10.1090/S0002-9947-2010-04887-1

Levi-flat hypersurfaces with real analytic boundary
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Trans. Amer. Math. Soc. 362 (2010), 6367-6380 Request permission

Abstract:

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 $\mathcal {O}(X)$-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.

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