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Local Normal Forms of Singular Levi-Flat Hypersurfaces

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Abstract

We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex quasihomogeneous polynomials with an isolated singularity at \(0\in \mathbb {C}^2\). This result extends previous results of Burns and Gong (Am J Math 121(1):23–53, 1999) and Fernández-Pérez (Ann Sc Norm Super Pisa Cl Sci (5) XIII:745–774, 2014). Furthermore, we prove the existence of two new rigid normal forms for singular real-analytic Levi-flat hypersurfaces which are preserved by a change of isochore coordinates, that is, a change of coordinates that preserves volume.

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Acknowledgements

The authors gratefully acknowledges the many helpful suggestions of Rogério Mol (UFMG) during the preparation of the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Minas Gerais, UFMG, Av. Antônio Carlos 6627, Belo Horizonte, MG, 31270-901, Brazil

    Arturo Fernández-Pérez

  2. Departamento de Matemática, Universidade Federal de Itajubá, Rua Irmã Ivone Drumond 200, Itabira, MG, 35903-087, Brazil

    Gustavo Marra

Authors
  1. Arturo Fernández-Pérez
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  2. Gustavo Marra
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Correspondence to Arturo Fernández-Pérez.

Additional information

This work is partially supported by CNPq Brazil Grant Number 427388/2016-3.

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Fernández-Pérez, A., Marra, G. Local Normal Forms of Singular Levi-Flat Hypersurfaces. J Geom Anal 29, 2776–2804 (2019). https://doi.org/10.1007/s12220-018-0094-3

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