Metric Properties of Domains in ℂn

Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 26)

Abstract

This article is the written form of a presentation given during the Workshop “Geometric Function Theory in higher dimension”. We present some developments in the classical field of classification of domains in \(\mathbb C^n\) and some related questions.

Keywords

Complex manifolds Automorphism groups Kobayashi hyperbolicity Horospheres D’Angelo finite type 

2010 Mathematics Subject Classification

32M99 32Q45 

Notes

Acknowledgements

The author would like to thank the organizers of the Workshop for the warm reception and the fruitful atmosphere. The author is partially supported by ERC ALKAGE.

References

  1. 1.
    Balogh, Z., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75, 504–533 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bedford, E., Fornaess, J.E.: A construction of peak functions on weakly pseudoconvex domains. Ann. Math. 107, 555–568 (1978)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bedford, E., Pinchuk, S.: Domains in \(\mathbb C^2\) with noncompact holomorphic automorphism groups. Math. Sb. 135, 147–157 (1988)Google Scholar
  4. 4.
    Bedford, E., Pinchuk, S.: Convex domains with noncompact automorphism groups. Math. Sb. 185, 3–26 (1994)MATHGoogle Scholar
  5. 5.
    Berteloot, F.: Characterization of models in \(\mathbb C^2\) by their automorphism groups. Int. J. Math. 5, 619–634 (1994)Google Scholar
  6. 6.
    Berteloot, F.: Méthodes de changement d’échelles en analyse complexe. Ann. Fac. Sci. Toulouse 15, 427–483(2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bracci, F., Gaussier, H.: Horosphere topology. ArXiv:1605.04119Google Scholar
  8. 8.
    Bracci, F., Gaussier, H.: A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in \(\mathbb C^n\). Math. Ann. doi:10.1007/s00208-017-1581-8Google Scholar
  9. 9.
    Bracci, F., Saracco, A.: Hyperbolicity in unbounded convex domains. Forum Math. 21, 815–825 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Byun, J., Gaussier, H., Lee, K.H.: On the automorphism group of strongly pseudoconvex domains in almost complex manifolds. Ann. Inst. Fourier 59, 291–310 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chern, S.S., Moser, J.: Real hypersurfaces in complex manifolds. Acta math. 133, 219–271 (1974)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    D’Angelo, J.P.: Finite type conditions for real hypersurfaces. J. Differ. Geom. 14, 59–66 (1979)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. 115, 615–637 (1982)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D’Angelo, J.P., Kohn, J.: Subelliptic estimates and finite type. In: Several Complex Variables. MSRI Publications, vol. 37, pp. 199–232. Cambridge University Press, Cambridge (1999)Google Scholar
  15. 15.
    Demailly, J.P.: Recent progress towards the Kobayashi and Green-Griffiths-Lang conjectures (expanded version of talks given at the 16th Takagi lectures). https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/takagi16_jpd.pdf
  16. 16.
    Gaussier, H.: Tautness and complete hyperbolicity of domains in \(\mathbb C^n\). Proc. Am. Math. Soc. 127, 105–116 (1999)Google Scholar
  17. 17.
    Gaussier, H., Kim, K.T., Krantz, S.: A note on the Wong-Rosay Theorem in complex manifolds. Complex Var. Theory Appl. 47, 761–768 (2002)MathSciNetMATHGoogle Scholar
  18. 18.
    Gaussier, H., Seshadri, H.: On the Gromov hyperbolicity of convex domains in \(\mathbb C^n\). To appear in CMFT. ArXiv:1312.0368Google Scholar
  19. 19.
    Harz, T., Shcherbina, N., Tomassini, G.: On defining functions and cores for unbounded domains. I. Math. Z. 286, 987–1002 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Isaev, A.: Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds. Lecture Notes in Mathematics, vol. 902.Google Scholar
  21. 21.
    Kim, K.T.: On the automorphism groups of convex domains in \(\mathbb C^n\). Adv. Geom. 4, 33–40 (2004)Google Scholar
  22. 22.
    Kobayashi, S.: Hyperbolic Complex Spaces. Grundlehren der mathematischen Wissenschaften, vol. 318. Springer, Berlin (1998)Google Scholar
  23. 23.
    Nikolov, N., Thomas, P., Trybula, M.: Gromov (non)hyperbolicity of certain domains in \(\mathbb C^n\). Forum Math. 28, 783–794 (2016)Google Scholar
  24. 24.
    Pinchuk, S.: The scaling method and holomorphic mappings, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), vol. 52, pp. 151–161, American Mathematical Society, Providence (1991)Google Scholar
  25. 25.
    Poincaré, H.: Les fonctions analytiques de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo, pp. 185–220 (1907)Google Scholar
  26. 26.
    Rosay, J.P.: Sur une caractérisation de la boule parmi les domaines de \(\mathbb C^n\) par son groupe d’automorphismes. Ann. Inst. Fourier 29, 91–97 (1979)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wong, B.: Characterization of the unit ball in \(\mathbb C^n\) by its automorphism group. Invent. Math. 41, 253–257 (1977)Google Scholar
  28. 28.
    Zimmer, A.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365, 142–198 (2016)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zimmer, A.: Characterizing domains by the limit set of their automorphism group. Adv. Math. 308, 438–482 (2017)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zimmer, A.: Gromov hyperbolicity, the Kobayashi metric, and C-convex sets. Trans. Am. Math. Soc. 369, 8437–8456 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Université Grenoble AlpesCNRSGrenobleFrance

Personalised recommendations