On the automorphism group of strongly pseudoconvex domains in almost complex manifolds

Byun, Jisoo; Gaussier, Hervé; Lee, Kang-Hyurk

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 291-310.

Contrairement au cas intégrable, il existe une infinité de variétés presque complexes homogènes, non intégrables, strictement pseudoconvexes en tout point de leur bord. De telles variétés sont équivalentes au demi-espace de Siegel muni d’une structure presque complexe linéaire.

Nous démontrons qu’il n’existe pas de représentation relativement compacte, strictement pseudoconvexe, de ces variétés. Enfin, nous étudions la semi-continuité du groupe des automorphismes de certaines variétés presque complexes hyperboliques, strictement pseudoconvexes, par déformation de la structure.

In contrast with the integrable case there exist infinitely many non-integrable homogeneous almost complex manifolds which are strongly pseudoconvex at each boundary point. All such manifolds are equivalent to the Siegel half space endowed with some linear almost complex structure.

We prove that there is no relatively compact strongly pseudoconvex representation of these manifolds. Finally we study the upper semi-continuity of the automorphism group of some hyperbolic strongly pseudoconvex almost complex manifolds under deformation of the structure.

Classification : 32G05, 32H02, 32T15, 53C15
Keywords: Automorphism groups, strongly pseudoconvex domains, almost complex manifolds
Mot clés : groupe d’automorphismes, domaines strictement pseudoconvexes, variétés presque complexes

[1] Bedford, E.; Pinchuk, S. I. Domains in ${\mathbf{C}}^2$ with noncompact groups of holomorphic automorphisms, Mat. Sb. (N.S.), Volume 135(177) (1988) no. 2, p. 147-157, 271 | MR | Zbl

[2] Brody, Robert Compact manifolds in hyperbolicity, Trans. Amer. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[3] Fridman, Buma L.; Ma, Daowei Perturbation of domains and automorphism groups, J. Korean Math. Soc., Volume 40 (2003) no. 3, pp. 487-501 | DOI | MR | Zbl

[4] Fridman, Buma L.; Ma, Daowei; Poletsky, Evgeny A. Upper semicontinuity of the dimensions of automorphism groups of domains in ${ℂ}^N$, Amer. J. Math., Volume 125 (2003) no. 2, pp. 289-299 | DOI | MR | Zbl

[5] Gaussier, Hervé; Kim, Kang-Tae Compactness of certain families of pseudo-holomorphic mappings into ${ℂ}^n$, Internat. J. Math., Volume 15 (2004) no. 1, pp. 1-12 | DOI | MR | Zbl

[6] Gaussier, Hervé; Kim, Kang-Tae; Krantz, Steven G. A note on the Wong-Rosay theorem in complex manifolds, Complex Var. Theory Appl., Volume 47 (2002) no. 9, pp. 761-768 | MR | Zbl

[7] Gaussier, Hervé; Sukhov, Alexandre Estimates of the Kobayashi-Royden metric in almost complex manifolds, Bull. Soc. Math. France, Volume 133 (2005) no. 2, pp. 259-273 | Numdam | MR | Zbl

[8] Gaussier, Hervé; Sukhov, Alexandre On the geometry of model almost complex manifolds with boundary, Math. Z., Volume 254 (2006) no. 3, pp. 567-589 | DOI | MR | Zbl

[9] Greene, Robert E.; Krantz, Steven G. The automorphism groups of strongly pseudoconvex domains, Math. Ann., Volume 261 (1982) no. 4, pp. 425-446 | DOI | MR | Zbl

[10] Greene, Robert E.; Krantz, Steven G. Normal families and the semicontinuity of isometry and automorphism groups, Math. Z., Volume 190 (1985) no. 4, pp. 455-467 | DOI | MR | Zbl

[11] Gromov, M. Pseudoholomorphic curves in symplectic manifolds, Invent. Math., Volume 82 (1985) no. 2, pp. 307-347 | DOI | MR | Zbl

[12] Hamilton, Richard S. Deformation of complex structures on manifolds with boundary. I. The stable case, J. Differential Geometry, Volume 12 (1977) no. 1, pp. 1-45 | MR | Zbl

[13] Ivashkovich, Sergey; Pinchuk, Sergey; Rosay, Jean-Pierre Upper semi-continuity of the Kobayashi-Royden pseudo-norm, a counterexample for Hölderian almost complex structures, Ark. Mat., Volume 43 (2005) no. 2, pp. 395-401 | DOI | MR | Zbl

[14] Ivashkovich, Sergey; Rosay, Jean-Pierre Schwarz-type lemmas for solutions of $\overline{\partial }$-inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 7, p. 2387-2435 (2005) | DOI | Numdam | MR | Zbl

[15] Kobayashi, Shoshichi Problems related to hyperbolicity of almost complex structures, Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday (Adv. Stud. Pure Math.), Volume 42, Math. Soc. Japan, Tokyo, 2004, pp. 141-146 | MR | Zbl

[16] Kruglikov, Boris Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm, C. R. Math. Acad. Sci. Paris, Volume 338 (2004) no. 4, pp. 295-299 | MR | Zbl

[17] Kruglikov, Boris S.; Overholt, Marius Pseudoholomorphic mappings and Kobayashi hyperbolicity, Differential Geom. Appl., Volume 11 (1999) no. 3, pp. 265-277 | DOI | MR | Zbl

[18] Landau, E. Collected works | Zbl

[19] Lee, Kang-Hyurk Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J., Volume 54 (2006) no. 1, pp. 179-205 | DOI | MR | Zbl

[20] Lee, Kang-Hyurk Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math., Volume 623 (2008), pp. 123-160 (to appear) | DOI | MR

[21] Lohwater, A. J.; Pommerenke, Ch. On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I (1973) no. 550, pp. 12 | MR | Zbl

[22] Nijenhuis, Albert; Woolf, William B. Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2), Volume 77 (1963), pp. 424-489 | DOI | MR | Zbl

[23] Pinchuk, Sergey The scaling method and holomorphic mappings, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) (Proc. Sympos. Pure Math.), Volume 52, Amer. Math. Soc., Providence, RI, 1991, pp. 151-161 | MR | Zbl

[24] Rosay, Jean-Pierre Sur une caractérisation de la boule parmi les domaines de ${\mathbf{C}}^n$ par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble), Volume 29 (1979) no. 4, pp. ix, 91-97 | DOI | Numdam | MR | Zbl

[25] Royden, H. L. Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Springer, Berlin, 1971, p. 125-137. Lecture Notes in Math., Vol. 185 | MR | Zbl

[26] Sikorav, Jean-Claude Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry (Progr. Math.), Volume 117, Birkhäuser, Basel, 1994, pp. 165-189 | MR

[27] Wong, B. Characterization of the unit ball in ${\mathbf{C}}^n$ by its automorphism group, Invent. Math., Volume 41 (1977) no. 3, pp. 253-257 | DOI | MR | Zbl

[28] Zalcman, Lawrence A heuristic principle in complex function theory, Amer. Math. Monthly, Volume 82 (1975) no. 8, pp. 813-817 | DOI | MR | Zbl