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arXiv · 1105.4417

Complex Plateau problem: old and new results and prospects

Pierre Dolbeault
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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.

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Pierre Dolbeault. 2011-05-23. Complex Plateau problem: old and new results and prospects. https://arxiv.org/abs/1105.4417

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