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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Ahern, P., Gong, X.: Real analytic manifolds in \({\mathbb{C}}^n\) with parabolic complex tangents along a submanifold of codimension one. Ann. Fac. Sci. Toulouse Math. 18(1), 1–64 (2009)
Bedford, E., Gaveau, B.: Envelopes of holomorphy of certain 2-spheres in \({\mathbf{C}}^2\). Amer. J. Math. 105, 975–1009 (1983)
Burcea, V.: A normal form for a submanifold \(M\subset {\mathbb{C}}^{N+1}\) of codimension 2 near a flat CR singularity (2011) (preprint)
Burcea, V.: On a family of analytic discs attached to a real submanifold \(M\) in \({\mathbb{C}}^{N+1}\). arXiv:1201.4136 (2012) (preprint)
Bishop, E.: Differentiable manifolds in complex Euclidean space. Duke Math. J. 32, 1–21 (1965)
Cartan, É.: Sur les variétés pseudo-conformal des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. Pura Appl. 4(11), 17–90 (1932)
Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)
Coffman, A.: Unfolding CR singularities. Mem. Amer. Math. Soc. 205(962) (2010)
Coffman, A.: CR singularities of real fourfolds in \({\mathbb{C}}^3\). Illinois J. Math. 53(3), 939–981 (2009)
Dolbeault, P., Tomassini, G., Zaitsev, D.: On Levi-flat hypersyrfaces with prescribed boundary. Pure Appl. Math. Q. 6(3), 725–753 (2010)
Dolbeault, P., Tomassini, G., Zaitsev, D.: Boundary problem for Levi flat graphs (2011) (to appear in Indiana Math. Journal) (preprint)
Gong, X.: On the convergence of normalilations of real analytic surfaces near hyperbolic complex tangents. Comment. Math. Helv. 69(4), 549–574 (1994)
Gong, X.: Normal forms of real surfaces under unimodular transformations near elliptic complex tangents. Duke Math. J. 74(1), 145–157 (1994)
Gong, X.: Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics. Indiana Univ. Math. J. 53(1), 83–95 (2004)
Forstneri, F.: Complex tangents of real surfaces in complex surfaces. Duke Math. J. 67(2), 353–376 (1992)
Huang, X.: On an n-manifold in \({\mathbb{C}}^n\) near an elliptic complex tangent. J. Amer. Math. Soc. 11, 669–692 (1998)
Huang, X.: Geometric Analysis in Several Complex Variables, Ph.D. Thesis, Washington University in St. Louis. http://www.math.rutgers.edu/huangx/thesis-huang (1994)
Huang, X., Krantz, S.: On a problem of Moser. Duke Math. J. 78, 213–228 (1995)
Huang, X., Yin, W.: A Bishop surface with a vanishing Bishop invariant. Invent. Math. 176(3), 461–520 (2009)
Huang, X., Yin, W.: A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric. Int. Math. Res. Notices. (15), 2789–2828 (2009)
Huang, X., Yin, W.: Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60. http://math.rutgers.edu/huangx/huang-yin-flatness-09-2014_2) (2014) (preprint)
Huang, X., Yin, W.: Flattening of CR singular points and analyticity of the local hull of holomorphy. arXiv:1210.5146 (2012) (preprint)
Lebl, J.: Algebraic Levi-flat hypervarieties in complex projective space. J. Geom. Anal. 22, 410–432 (2012)
Lebl, J.: Extension of Levi-flat hypersurfaces past CR boundaries. Indiana Univ. Math. J. 57(2), 699–716 (2008)
Mir, N.: Convergence of formal embeddings between real-analytic hypersurfaces in codimension one. J. Differ.Geom. 62, 163–173 (2002)
Moser, J.: Analytic surfaces in \({\mathbb{C}}^2\) and their local hull of holomorphy. Ann AcademiæFennicae Series A I Math 10, 397–410 (1985)
Moser, J., Webster, S.: Normal forms for real surfaces in \({\mathbb{C}}^2\) near complex tangents and hyperbolic surface transformations. Acta Math. 150, 255–296 (1983)
Kenig, C., Webster, S.: The local hull of holomorphy of a surface in the space of two complex variables. Invent. Math. 67, 1–21 (1982)
Kenig, C., Webster, S.: On the hull of holomorphy of an n-manifold in \({\mathbb{C}}^n\), Annali Scoula Norm. Sup. de Pisa IV, 11(2), 261–280 (1984)
Kohn, J.: Boundary behavior of \(\overline{\partial }\) on weakly pseudo-convex manifolds of dimension two, 6, 523–542 (1972)
Stolovitch, L.: Family of intersecting totally real manifolds of \(({\mathbb{C}}^n,0)\) and CR-singularities, preprint. arXiv:math/0506052
Trepreau, J.P.: Sur le prolongement holomorphe des fonctions CR definies sur une hypersurface reele de classe \({\mathbb{C}}^2\) dans \({\mathbb{C}}^n\). Invent. Math. 83, 583–592 (1986)
Tumanov, A.E.: Extension of CR-functions into a wedge. Mat. Sb. 181(7), 951–964 (1990) (translation in Math. USSR-Sb. 70 (1991), no. 2, 385–398)
Zaitsev, D.: Formal and finite order equivalences. Math. Z. 269(3–4), 687–696 (2011)
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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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