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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. |
