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

Jisoo Byun[1]; Hervé Gaussier[2]; Kang-Hyurk Lee[3]

  • [1] Department of Mathematics POSTECH Pohang, 790-784 (Korea)
  • [2] CMI 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France)
  • [3] School of Mathematics KIAS, Hoegiro 87 Dongdaemun-gu Seoul, 130-722 (Korea)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 291-310
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Byun, Jisoo, Gaussier, Hervé, and Lee, Kang-Hyurk. "On the automorphism group of strongly pseudoconvex domains in almost complex manifolds." Annales de l’institut Fourier 59.1 (2009): 291-310. <http://eudml.org/doc/10393>.

@article{Byun2009,
abstract = {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.},
affiliation = {Department of Mathematics POSTECH Pohang, 790-784 (Korea); CMI 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France); School of Mathematics KIAS, Hoegiro 87 Dongdaemun-gu Seoul, 130-722 (Korea)},
author = {Byun, Jisoo, Gaussier, Hervé, Lee, Kang-Hyurk},
journal = {Annales de l’institut Fourier},
keywords = {Automorphism groups; strongly pseudoconvex domains; almost complex manifolds; non-integrable deformations; automorphism groups},
language = {eng},
number = {1},
pages = {291-310},
publisher = {Association des Annales de l’institut Fourier},
title = {On the automorphism group of strongly pseudoconvex domains in almost complex manifolds},
url = {http://eudml.org/doc/10393},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Byun, Jisoo
AU - Gaussier, Hervé
AU - Lee, Kang-Hyurk
TI - On the automorphism group of strongly pseudoconvex domains in almost complex manifolds
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 291
EP - 310
AB - 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.
LA - eng
KW - Automorphism groups; strongly pseudoconvex domains; almost complex manifolds; non-integrable deformations; automorphism groups
UR - http://eudml.org/doc/10393
ER -

References

top
  1. E. Bedford, S. I. Pinchuk, Domains in C 2 with noncompact groups of holomorphic automorphisms, Mat. Sb. (N.S.) 135(177) (1988), 147-157, 271 Zbl0643.32011MR937803
  2. Robert Brody, Compact manifolds in hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213-219 Zbl0416.32013MR470252
  3. Buma L. Fridman, Daowei Ma, Perturbation of domains and automorphism groups, J. Korean Math. Soc. 40 (2003), 487-501 Zbl1039.32031MR1973914
  4. Buma L. Fridman, Daowei Ma, Evgeny A. Poletsky, Upper semicontinuity of the dimensions of automorphism groups of domains in ℂ N , Amer. J. Math. 125 (2003), 289-299 Zbl1031.32019MR1963686
  5. Hervé Gaussier, Kang-Tae Kim, Compactness of certain families of pseudo-holomorphic mappings into ℂ n , Internat. J. Math. 15 (2004), 1-12 Zbl1058.32020MR2039209
  6. Hervé Gaussier, Kang-Tae Kim, Steven G. Krantz, A note on the Wong-Rosay theorem in complex manifolds, Complex Var. Theory Appl. 47 (2002), 761-768 Zbl1044.32019MR1925173
  7. Hervé Gaussier, Alexandre Sukhov, Estimates of the Kobayashi-Royden metric in almost complex manifolds, Bull. Soc. Math. France 133 (2005), 259-273 Zbl1083.32011MR2172267
  8. Hervé Gaussier, Alexandre Sukhov, On the geometry of model almost complex manifolds with boundary, Math. Z. 254 (2006), 567-589 Zbl1107.32009MR2244367
  9. Robert E. Greene, Steven G. Krantz, The automorphism groups of strongly pseudoconvex domains, Math. Ann. 261 (1982), 425-446 Zbl0531.32016MR682655
  10. Robert E. Greene, Steven G. Krantz, Normal families and the semicontinuity of isometry and automorphism groups, Math. Z. 190 (1985), 455-467 Zbl0584.58014MR808913
  11. M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347 Zbl0592.53025MR809718
  12. Richard S. Hamilton, Deformation of complex structures on manifolds with boundary. I. The stable case, J. Differential Geometry 12 (1977), 1-45 Zbl0394.32010MR477158
  13. Sergey Ivashkovich, Sergey Pinchuk, Jean-Pierre Rosay, Upper semi-continuity of the Kobayashi-Royden pseudo-norm, a counterexample for Hölderian almost complex structures, Ark. Mat. 43 (2005), 395-401 Zbl1091.32009MR2173959
  14. Sergey Ivashkovich, Jean-Pierre Rosay, Schwarz-type lemmas for solutions of ∂ ¯ -inequalities and complete hyperbolicity of almost complex manifolds, Ann. Inst. Fourier (Grenoble) 54 (2004), 2387-2435 (2005) Zbl1072.32007MR2139698
  15. Shoshichi Kobayashi, Problems related to hyperbolicity of almost complex structures, Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday 42 (2004), 141-146, Math. Soc. Japan, Tokyo Zbl1079.32014MR2087047
  16. Boris Kruglikov, Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm, C. R. Math. Acad. Sci. Paris 338 (2004), 295-299 Zbl1041.32015MR2076499
  17. Boris S. Kruglikov, Marius Overholt, Pseudoholomorphic mappings and Kobayashi hyperbolicity, Differential Geom. Appl. 11 (1999), 265-277 Zbl0954.32019MR1726542
  18. E. Landau, Collected works Zbl0653.01019
  19. Kang-Hyurk Lee, Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J. 54 (2006), 179-205 Zbl1145.32014MR2214793
  20. Kang-Hyurk Lee, Strongly pseudoconvex homogeneous domains in almost complex manifolds, J. Reine Angew. Math. 623 (2008), 123-160 Zbl1160.32024MR2458042
  21. A. J. Lohwater, Ch. Pommerenke, On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I (1973) Zbl0275.30027MR338381
  22. Albert Nijenhuis, William B. Woolf, Some integration problems in almost-complex and complex manifolds, Ann. of Math. (2) 77 (1963), 424-489 Zbl0115.16103MR149505
  23. Sergey Pinchuk, The scaling method and holomorphic mappings, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) 52 (1991), 151-161, Amer. Math. Soc., Providence, RI Zbl0744.32013MR1128522
  24. Jean-Pierre Rosay, Sur une caractérisation de la boule parmi les domaines de C n par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble) 29 (1979), ix, 91-97 Zbl0402.32001MR558590
  25. H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) (1971), 125-137. Lecture Notes in Math., Vol. 185, Springer, Berlin Zbl0218.32012MR304694
  26. Jean-Claude Sikorav, Some properties of holomorphic curves in almost complex manifolds, Holomorphic curves in symplectic geometry 117 (1994), 165-189, Birkhäuser, Basel MR1274929
  27. B. Wong, Characterization of the unit ball in C n by its automorphism group, Invent. Math. 41 (1977), 253-257 Zbl0385.32016MR492401
  28. Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813-817 Zbl0315.30036MR379852

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.