Complex Plateau problem: old and new results and prospects

TL;DR

This paper reviews the complex Plateau problem in Hermitian manifolds, discussing classical and new results on extending real manifolds into complex subvarieties, and explores Levi-flat hypersurfaces with prescribed boundaries.

Contribution

It provides a comprehensive overview of old and new results on the complex Plateau problem and proposes extensions to real parametric problems for constructing Levi-flat hypersurfaces.

Findings

Recalled minimality properties of complex and Levi-flat subvarieties.

Summarized known results in complex and projective spaces.

Proposed solutions and constructions for Levi-flat hypersurfaces with prescribed boundaries.

Abstract

The complex Plateau problem is analogous, in a Hermitian complex manifold, to the classical Plateau problem in 3 dimensional real space: it is a geometrical problem of extension of a closed real manifold into a complex analytic subvariety, or into a Levi-flat subvariety. Minimality of complex analytic subvarieties and analogous properties of Levi-flat subvarieties, in K\"ahler manifolds, are recalled or given. Known results in n dimensional complex and projective complex spaces are recalled. Extensions to real parametric problems are solved or proposed, leading to the construction of Levi-flat hypersurfaces with prescribed boundary in some complex manifolds.