On an 𝑛-manifold in 𝐂ⁿ near an elliptic complex tangent

Huang, Xiaojun¹

¹ Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Journal of the American Mathematical Society, Tome 11 (1998) no. 3, pp. 669-692

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-98-00265-3

Bibliographie

[1] Alexander, H. Gromov’s method and Bennequin’s problem Invent. Math. 1996 135 148

[2] Alexander, H. Linking and holomorphic hulls J. Differential Geom. 1993 151 160

[3] Baouendi, M. S., Rothschild, L. P., Trã©Preau, J.-M. On the geometry of analytic discs attached to real manifolds J. Differential Geom. 1994 379 405

[4] Saxena, R. K., Hussain, Z. Generating functions for Jacobi polynomials Univ. Nac. Tucumán Rev. Ser. A 1976

[5] Bedford, Eric, Klingenberg, Wilhelm On the envelope of holomorphy of a 2-sphere in 𝐶² J. Amer. Math. Soc. 1991 623 646

[6] Bishop, Errett Differentiable manifolds in complex Euclidean space Duke Math. J. 1965 1 21

[7] Boggess, Albert CR manifolds and the tangential Cauchy-Riemann complex 1991

[8] Debiard, Amã©Dã©E, Gaveau, Bernard Problème de Dirichlet pour l’équation de Lévi Bull. Sci. Math. (2) 1978 369 386

[9] Eliashberg, Yakov Filling by holomorphic discs and its applications 1990 45 67

[10] Deimling, Klaus Nonlinear functional analysis 1985

[11] Duval, Julien, Sibony, Nessim Polynomial convexity, rational convexity, and currents Duke Math. J. 1995 487 513

[12] Forstneriä, Franc Analytic disks with boundaries in a maximal real submanifold of 𝐶² Ann. Inst. Fourier (Grenoble) 1987 1 44

[13] Forstneriä, Franc Complex tangents of real surfaces in complex surfaces Duke Math. J. 1992 353 376

[14] Gromov, M. Pseudo holomorphic curves in symplectic manifolds Invent. Math. 1985 307 347

[15] Hill, C. Denson, Taiani, Geraldine Families of analytic discs in 𝐶ⁿ with boundaries on a prescribed CR submanifold Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1978 327 380

[16] Huang, Xiao Jun, Krantz, Steven G. On a problem of Moser Duke Math. J. 1995 213 228

[17] Hunt, L. R. The local envelope of holomorphy of an 𝑛-manifold in 𝐶ⁿ Boll. Un. Mat. Ital. (4) 1971 12 35

[18] Hunt, L. R., Wells, R. O., Jr. The envelope of holomorphy of a two-manifold in C2 Rice Univ. Stud. 1970

[19] Kenig, Carlos E., Webster, Sidney M. The local hull of holomorphy of a surface in the space of two complex variables Invent. Math. 1982 1 21

[20] Kenig, Carlos E., Webster, Sidney M. On the hull of holomorphy of an 𝑛-manifold in 𝐶ⁿ Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1984 261 280

[21] Lempert, Lã¡Szlã³ La métrique de Kobayashi et la représentation des domaines sur la boule Bull. Soc. Math. France 1981 427 474

[22] Moser, Jã¼Rgen Analytic surfaces in 𝐶² and their local hull of holomorphy Ann. Acad. Sci. Fenn. Ser. A I Math. 1985 397 410

[23] Sasaki, Takeshi Classification of invariant complex structures on 𝑆𝐿(3,𝑅) Kumamoto J. Sci. (Math.) 1982 59 72

[24] Bland, John, Epstein, C. L. Embeddable CR-structures and deformations of pseudoconvex surfaces. I. Formal deformations J. Algebraic Geom. 1996 277 368

[25] Trã©Preau, J.-M. Sur le prolongement holomorphe des fonctions C-R défines sur une hypersurface réelle de classe 𝐶² dans 𝐶ⁿ Invent. Math. 1986 583 592

[26] Tumanov, A. E. Extension of CR-functions into a wedge from a manifold of finite type Mat. Sb. (N.S.) 1988 128 139