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Flattening of CR singular points and analyticity of the local hull of holomorphy I

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Abstract

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in \({\mathbb C}^n\) with \(n\ge 3\) near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

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Acknowledgments

The paper was more or less completed in the summer of 2011 when the first author was visiting Wuhan University. The first author would like to thank the School of Mathematics and Statistics, Wuhan University for the hospitality during his stay. Part of the work in the paper had been done while the second author was taking a year long visit at Rutgers University at New Brunswick in 2009. The second author likes to thank this institute for the hospitality during his stay. The first author would like to thank Dima Zaistev for his very stimulating and helpful conversations related to this work. The second author also likes very much to thank Nordine Mir for his many helps both in his mathematics and in other arrangements during his stay at the University of Rouen, through a European Union postdoctoral fellowship.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Xiaojun Huang & Wanke Yin

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    Xiaojun Huang

  3. Laboratoire de Mathématiques Raphaël Salem, Université de Rouen, 76801, Saint-Etienne-du-Rouvray, France

    Wanke Yin

Authors
  1. Xiaojun Huang
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  2. Wanke Yin
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Correspondence to Xiaojun Huang.

Additional information

X. Huang supported in part by NSF-1363418. W. Yin supported in part by FANEDD-201117, ANR-09-BLAN-0422, NSFC-10901123 and NSFC-11271291.

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Huang, X., Yin, W. Flattening of CR singular points and analyticity of the local hull of holomorphy I. Math. Ann. 365, 381–399 (2016). https://doi.org/10.1007/s00208-015-1228-6

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  • DOI: https://doi.org/10.1007/s00208-015-1228-6

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