Bounding smooth Levi-flat hypersurfaces in a Stein manifold

TL;DR

This paper proves the existence of Levi-flat hypersurfaces attached to certain submanifolds in Stein manifolds, solving a longstanding problem and revealing complex interactions with symplectic geometry and foliation theory.

Contribution

It establishes conditions under which smooth Levi-flat hypersurfaces exist attached to real codimension two submanifolds in Stein manifolds, addressing a question from Bishop's 1965 problem.

Findings

Existence of Levi-flat hypersurfaces under CR non-minimal condition

Boundaries of strongly pseudoconvex domains support Levi-flat hypersurfaces

Resolution of a classical open problem in complex analysis

Abstract

This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR singularities, a research direction originated from a fundamental and classical paper of E. Bishop. Earlier works along these lines include those by many prominent mathematicians working both on complex analysis and geometry. We prove that a compact smooth (or, real analytic) real codimension two submanifold $M$, that is contained in the boundary of a smoothly bounded strongly pseudoconvex domain, with a natural and necessary condition called CR non-minimal condition at CR points and with two elliptic CR singular points bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively) Levi-flat hypersurface $\widehat{M}$. This answers a well-known question left open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of a problem already asked by Bishop in 1965. Our study here reveals an intricate interaction of several complex analysis with other fields such as symplectic geometry and foliation theory.