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

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- by Xiaojun Huang PDF
- J. Amer. Math. Soc.
**11**(1998), 669-692 Request permission

## 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

- H. Alexander,
*Gromov’s method and Bennequin’s problem*, Invent. Math.**125**(1996), no. 1, 135–148. MR**1389963**, DOI 10.1007/s002220050071 - H. Alexander,
*Linking and holomorphic hulls*, J. Differential Geom.**38**(1993), no. 1, 151–160. MR**1231705**, DOI 10.4310/jdg/1214454098 - M. S. Baouendi, L. P. Rothschild, and J.-M. Trépreau,
*On the geometry of analytic discs attached to real manifolds*, J. Differential Geom.**39**(1994), no. 2, 379–405. MR**1267896**, DOI 10.4310/jdg/1214454877 - R. K. Saxena and Z. Hussain,
*Generating functions for Jacobi polynomials*, Univ. Nac. Tucumán Rev. Ser. A**26**(1976), no. 1-2, 45–48 (1981). MR**727616** - Eric Bedford and Wilhelm Klingenberg,
*On the envelope of holomorphy of a $2$-sphere in $\textbf {C}^2$*, J. Amer. Math. Soc.**4**(1991), no. 3, 623–646. MR**1094437**, DOI 10.1090/S0894-0347-1991-1094437-0 - Errett Bishop,
*Differentiable manifolds in complex Euclidean space*, Duke Math. J.**32**(1965), 1–21. MR**200476** - Albert Boggess,
*CR manifolds and the tangential Cauchy-Riemann complex*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR**1211412** - Amédée Debiard and Bernard Gaveau,
*Problème de Dirichlet pour l’équation de Lévi*, Bull. Sci. Math. (2)**102**(1978), no. 4, 369–386 (French, with English summary). MR**517769** - Yakov Eliashberg,
*Filling by holomorphic discs and its applications*, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR**1171908** - Klaus Deimling,
*Nonlinear functional analysis*, Springer-Verlag, Berlin, 1985. MR**787404**, DOI 10.1007/978-3-662-00547-7 - Julien Duval and Nessim Sibony,
*Polynomial convexity, rational convexity, and currents*, Duke Math. J.**79**(1995), no. 2, 487–513. MR**1344768**, DOI 10.1215/S0012-7094-95-07912-5 - Franc Forstnerič,
*Analytic disks with boundaries in a maximal real submanifold of $\textbf {C}^2$*, Ann. Inst. Fourier (Grenoble)**37**(1987), no. 1, 1–44 (English, with French summary). MR**894560**, DOI 10.5802/aif.1076 - Franc Forstnerič,
*Complex tangents of real surfaces in complex surfaces*, Duke Math. J.**67**(1992), no. 2, 353–376. MR**1177310**, DOI 10.1215/S0012-7094-92-06713-5 - M. Gromov,
*Pseudo holomorphic curves in symplectic manifolds*, Invent. Math.**82**(1985), no. 2, 307–347. MR**809718**, DOI 10.1007/BF01388806 - C. Denson Hill and Geraldine Taiani,
*Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**5**(1978), no. 2, 327–380. MR**501906** - X. Huang,
*Geometric Analysis in Several Complex Variables*, Ph. D. Thesis, Washington University in St. Louis, August, 1994 (Chapter 4). - Xiao Jun Huang and Steven G. Krantz,
*On a problem of Moser*, Duke Math. J.**78**(1995), no. 1, 213–228. MR**1328757**, DOI 10.1215/S0012-7094-95-07809-0 - L. R. Hunt,
*The local envelope of holomorphy of an $n$-manifold in $C^{n}$*, Boll. Un. Mat. Ital. (4)**4**(1971), 12–35. MR**0294690** - L. R. Hunt and R. O. Wells Jr.,
*The envelope of holomorphy of a two-manifold in C2*, Rice Univ. Stud.**56**(1970), no. 2, 51–62 (1971). MR**271396** - Carlos E. Kenig and Sidney M. Webster,
*The local hull of holomorphy of a surface in the space of two complex variables*, Invent. Math.**67**(1982), no. 1, 1–21. MR**664323**, DOI 10.1007/BF01393370 - Carlos E. Kenig and Sidney M. Webster,
*On the hull of holomorphy of an $n$-manifold in $\textbf {C}^n$*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**11**(1984), no. 2, 261–280. MR**764946** - László Lempert,
*La métrique de Kobayashi et la représentation des domaines sur la boule*, Bull. Soc. Math. France**109**(1981), no. 4, 427–474 (French, with English summary). MR**660145**, DOI 10.24033/bsmf.1948 - Jürgen Moser,
*Analytic surfaces in $\textbf {C}^2$ and their local hull of holomorphy*, Ann. Acad. Sci. Fenn. Ser. A I Math.**10**(1985), 397–410. MR**802502**, DOI 10.5186/aasfm.1985.1044 - Takeshi Sasaki,
*Classification of invariant complex structures on $\textrm {SL}(3,\,\textbf {R})$*, Kumamoto J. Sci. (Math.)**15**(1982), no. 1, 59–72. MR**671021** - John Bland and C. L. Epstein,
*Embeddable CR-structures and deformations of pseudoconvex surfaces. I. Formal deformations*, J. Algebraic Geom.**5**(1996), no. 2, 277–368. MR**1374711** - J.-M. Trépreau,
*Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe $C^2$ dans $\textbf {C}^n$*, Invent. Math.**83**(1986), no. 3, 583–592 (French). MR**827369**, DOI 10.1007/BF01394424 - A. E. Tumanov,
*Extension of CR-functions into a wedge from a manifold of finite type*, Mat. Sb. (N.S.)**136(178)**(1988), no. 1, 128–139 (Russian); English transl., Math. USSR-Sb.**64**(1989), no. 1, 129–140. MR**945904**, DOI 10.1070/SM1989v064n01ABEH003298

## Additional Information

**Xiaojun Huang**- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- Email: huangx@math.rutgers.edu
- 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
- © Copyright 1998 American Mathematical Society
- 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