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On the Levi-flat Plateau problem

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Abstract

We solve the Levi-flat Plateau problem in the following case. Let \(M \subset {\mathbb {C}}^{n+1}\), \(n \ge 2\), be a connected compact real-analytic codimension-two submanifold with only nondegenerate CR singularities. Suppose M is a diffeomorphic image via a real-analytic CR map of a real-analytic hypersurface in \({\mathbb {C}}^n \times {\mathbb {R}}\) with only nondegenerate CR singularities. Then there exists a unique compact real-analytic Levi-flat hypersurface, nonsingular except possibly for self-intersections, with boundary M. We also study boundary regularity of CR automorphisms of domains in \({\mathbb {C}}^n \times {\mathbb {R}}\).

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Correspondence to Jiří Lebl.

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Lebl, J., Noell, A. & Ravisankar, S. On the Levi-flat Plateau problem. Complex Anal Synerg 6, 3 (2020) doi:10.1007/s40627-019-0040-6

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Keywords

  • Levi-flat Plateau problem
  • CR singular
  • Holomorphic hull
  • CR function

Mathematics Subject Classification

  • 32V40 (Primary)
  • 32V25
  • 32E05 (Secondary)