Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

On an $n$-manifold in ${\mathbf{C}}^{n}$
near an elliptic complex tangent


Author: Xiaojun Huang
Journal: J. Amer. Math. Soc. 11 (1998), 669-692
MSC (1991): Primary 32F25, 32D05
DOI: https://doi.org/10.1090/S0894-0347-98-00265-3
MathSciNet review: 1603854
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the local biholomorphic property of a real $n$-manifold $M\subset\mathbf C^n$ near an elliptic complex tangent point $p\in M$. In particular, we are interested in the regularity and the unique disk-filling problem of the local hull of holomorphy $\widetilde{M}$ of $M$ near $p$, first considered in a paper of Bishop. When $M$ is a $C^{\infty}$-smooth submanifold, using a result established by Kenig-Webster, we show that near $p$, $\widetilde{M}$ is a smooth Levi-flat $(n+1)$-manifold with a neighborhood of $p$ in $M$ as part of its $C^{\infty}$ boundary. Moreover, near $p$, $\widetilde{M}$ is foliated by a family of disjoint embedded complex analytic disks. We also prove a uniqueness theorem for the analytic disks attached to $M$. This result was proved in the previous work of Kenig-Webster when $n=2$. When $M$ is real analytic, we show that $\widetilde{M}$ is real analytic with a neighborhood of $p$ in $M$ as part of its real analytic boundary. Equivalently, we prove the convergence of the formal solutions of a certain functional equation. When $n=2$ or when $n>2$ but the Bishop invariant does not vanish at the point under study, the analyticity was then previously obtained in the work of Moser-Webster, Moser, and in the author's joint work with Krantz.


References [Enhancements On Off] (What's this?)

  • [ALX1] H. Alexander, Gromov's method and Bennequin's problem, Invent. Math. 125 (1996), no. 1, 135-148. MR 97j:32007
  • [ALX2] H. Alexander, Linking and holomorphic hulls, J. Diff. Geom. 38 (1993), no. 1, 151-160. MR 94g:32013
  • [BRT] S. Baouendi, L. Rothschild and J-M. Trepreau, On the geometry of analytic disks attached to real manifolds, J. of Diff. Geom. 39 (1994), 379-409. MR 95a:32026
  • [BG] E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in ${\mathbf C}^{2}$, Amer. J. Math. 105 (1983), 975-1009. MR 84k:33016
  • [BK] E. Bedford and W. Klingenberg, On the envelopes of holomorphy of a 2-sphere in ${\mathbf C}^{2}$, J. of AMS 4 (1991), 623-655. MR 92j:32034
  • [BIS] E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32(1965), 1-21. MR 34:369
  • [Bog] A. Boggess, CR Manifold and the Tangential Cauchy-Riemann Complex, CRC Press, Boca Raton, Ann Arbor Boston, 1991. MR 94e:32035
  • [DG] A. Debiard and B. Gaveau, Problèm de Dirichlet pour L'équation de Lévi, Bull. Sc. Math. 102 (1978), 369-386. MR 80f:32010
  • [ELI] Y. Eliashberg, Filling by holomorphic discs and its applications, Geometry of Low Dimensional Manifolds, London Math. Soc. Lecture Notes, Vol. 151. MR 93g:53060
  • [Dei] K. Deimling, Non-Linear Functional Analysis, Springer-Verlag, 1985. MR 86j:47001
  • [DS] J. Duval and N. Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487-513. MR 96f:32016
  • [FO1] F. Forstneric, Analytic disks with boundaries in a maximal real submanifold of ${\mathbf{C}}^{2}$, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 1-44. MR 88j:32019
  • [FO2] F. Forstneric, Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), no. 2, 353-376. MR 93g:32025
  • [GRO] M. Gromov, Pseudo holomorphic curves in symplectic geometry, Invent Math. 82 (1985), 307-347. MR 87j:53053
  • [DH] C. D. Hill and G. Taiani, Families of analytic disks in ${\mathbf{C}}^{n}$ with boundaries on a prescribed CR submanifold, Annali Scoula Norm. Sup. Pisa 5 (1978), 327-380. MR 80c:32023
  • [HU] X. Huang, Geometric Analysis in Several Complex Variables, Ph. D. Thesis, Washington University in St. Louis, August, 1994 (Chapter 4).
  • [HK] X. Huang and S. Krantz, On a problem of Moser, Duke Math. Journal 78 (1995), 213-228. MR 96f:32026
  • [HT] L. R. Hunt, The local envelope of holomorphy of a two-manifold in $\mathbf{C}^{2}$, Boll. Un. Math. Ital. 4, 12-35 (1971) MR 45:3758
  • [HW] L. R. Hunt and R. O. Wells, The envelope of holomorphy of a two-manifold in $\mathbf{C}^{2}$, Prof. Conf. on Complex Analysis, Rice Univ. 1969, Rice Univ. Studies 56, 51-62(1970). MR 42:6279
  • [KW1] C. Kenig and S. Webster, The local hull of holomorphy of a surface in the space of two complex variables, Invent. Math. 67(1982), 1-21. MR 84c:32014
  • [KW2] C. Kenig and S. Webster, On the hull of holomorphy of an $n$-manifold in ${\mathbf{C}}^{n}$, Annali Scoula Norm. Sup. de Pisa IV, 11 No. 2, 261-280(1984). MR 86d:32019
  • [LE] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981) 427-474. MR 84d:32036
  • [MOS] J. Moser, Analytic surfaces in ${\mathbf C}^{2}$ and their local hull of holomorphy, Annales Academiae Fennicae, Series A.I.Mathematica 10(1985), 397-410. MR 87c:32024
  • [MW] J. Moser and S. Webster, Normal forms for real surfaces in ${\mathbf C}^{2}$ near complex tangents and hyperbolic surface transformations, Acta Math. 150(1983), 255-296. MR 85c:32034
  • [ST] Z. Slodkowski and G. Tomassini, Geometric properties of solutions of the Levi curvature equation in ${\mathbf{C}}\sp{2}$, J. Funct. Anal. 138 (1996), no. 1, 188-212. MR 97c:32021
  • [TR] J.M. Trepreau, Sur le prolongement holomorphe des fonctions CR définies sur une hypersurface réele de la classe $C^{2}$ dans ${\mathbf{C}}^{n}$, Invent Math. 83 (1986), 583-592. MR 87f:32035
  • [TU] A. Tumanov, Extension of CR functions into a wedge from a manifold of finite type, Mat. Sb. 136 (1988) 128-139; English transl. in Math. USSR-Sb. 64 (1989), 129-140. MR 89m:32027

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 32F25, 32D05

Retrieve articles in all journals with MSC (1991): 32F25, 32D05


Additional Information

Xiaojun Huang
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Email: huangx@math.rutgers.edu

DOI: https://doi.org/10.1090/S0894-0347-98-00265-3
Received by editor(s): August 8, 1997
Received by editor(s) in revised form: February 9, 1998
Additional Notes: The author was supported in part by NSF DMS-9500881 and an NSF postdoctoral fellowship
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society