Levi-flat hypersurfaces with real analytic boundary

TL;DR

This paper proves that Levi-flat hypersurfaces with real analytic boundary in Stein manifolds are themselves real analytic under certain conditions, extending the boundary analyticity into the interior and providing new variants of Malgrange's theorem.

Contribution

The paper establishes the real analyticity of Levi-flat hypersurfaces with real analytic boundary in Stein manifolds, including cases with CR singularities, and introduces new variants of Malgrange's theorem for submanifolds and subanalytic sets.

Findings

Levi-flat hypersurfaces are real analytic if boundary CR singularities are controlled.

Real algebraic boundaries lead to algebraic Levi-flat hypersurfaces extending past the boundary.

New variants of Malgrange's theorem for submanifolds with boundary and subanalytic sets.

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.