On the set of complex points of a 2-sphere

Shcherbina, Nikolay

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 1, pp. 73-87.

Let $G$ be a strictly pseudoconvex domain in ${ℂ}^2$ with $C^{\infty }$-smooth boundary $\partial G$. Let $S$ be a 2-dimensional sphere embedded into $\partial G$. Denote by $ℰ$ the set of all complex points on $S$. We study how the structure of the set $ℰ$ depends on the smoothness of $S$.

Classification : 32T15, 32V40, 53D10

[1] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque 107-108 (1983), 87–161. | Numdam | MR

[2] E. Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. | MR | Zbl

[3] E. Bedford and W. Klingenberg, On the envelope of holomorphy of a 2-sphere in ${ℂ}^2$, J. Amer. Math. Soc. 4 (1991), 623–643. | MR | Zbl

[4] E. M. Chirka, Regularity of the boundaries of analytic sets, Mat. Sb. 117 (1982), 291–336; English transl. in Sb. Math. 45 (1983), 291–335. | MR

[5] Ya. 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, 45–67. | MR

[6] O. G. Eroshkin, On a topological property of the boundary of an analytic subset of a strictly pseudoconvex domain in ${ℂ}^2$, Mat. Zametki 49 (1991), 149–151; English transl. in Math. Notes 49 (1991), 546–547. | MR

[7] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307–347. | EuDML | MR | Zbl

[8] M. W. Hirsch, “Differential Topology”, Grad. Texts in Math., Vol. 33, Springer-Verlag, New York, 1976. | MR | Zbl

[9] B. Jöricke, Local polynomial hulls of discs near isolated parabolic points, Indiana Univ. Math. J. 46 (1997), 789–826. | MR | Zbl

[10] N. G. Kružilin, Two-dimensional spheres on the boundaries of pseudoconvex domains in ${ℂ}^2$, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 1194–1237; English transl. in Math. USSR Izv. 39 (1992), 1151–1187. | MR

[11] H. F. Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. Amer. Math. Soc. 172 (1972), 1–33. | MR | Zbl

[12] S. Nemirovski, Complex analysis and differential topology on complex surfaces, Uspekhi Mat. Nauk 54 (1999), 47–74; English transl. in Russian Math. Surveys 54 (1999), 729–752. | MR

[13] E. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton Univ. Press, Princeton, 1970. | MR | Zbl

[14] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514–517. | JFM | MR

[15] J. Wiegerinck, Local polynomially convex hulls at degenerated $CR$ singularities of surfaces in ${ℂ}^2$, Indiana Univ. Math. J. 44 (1995), 897–915. | MR | Zbl