Tohoku Mathematical Journal

Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel

Atsushi Yamamori

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In the study of the holomorphic automorphism groups, many researches have been carried out inside the category of bounded or hyperbolic domains. On the contrary to these cases, for unbounded non-hyperbolic cases, only a few results are known about the structure of the holomorphic automorphism groups. Main result of the present paper gives a class of unbounded non-hyperbolic Reinhardt domains with non-compact automorphism groups, Cartan's linearity theorem and explicit Bergman kernels. Moreover, a reformulation of Cartan's linearity theorem for finite volume Reinhardt domains is also given.

Article information

Source
Tohoku Math. J. (2), Volume 69, Number 2 (2017), 239-260.

Dates
First available in Project Euclid: 24 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1498269625

Digital Object Identifier
doi:10.2748/tmj/1498269625

Mathematical Reviews number (MathSciNet)
MR3682165

Zentralblatt MATH identifier
06775254

Subjects
Primary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]
Secondary: 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)

Keywords
Bergman kernel holomorphic automorphism group Reinhardt domain

Citation

Yamamori, Atsushi. Non-hyperbolic unbounded Reinhardt domains: non-compact automorphism group, Cartan's linearity theorem and explicit Bergman kernel. Tohoku Math. J. (2) 69 (2017), no. 2, 239--260. doi:10.2748/tmj/1498269625. https://projecteuclid.org/euclid.tmj/1498269625


Export citation

References

  • K. Azukawa, Square-integrable holomorphic functions on a circular domains in $C^n$, Tohoku Math. J. (2) 37 (1985), no. 1, 15–26.
  • E. Bedford and S. Pinchuk, Domains in $C^{n+1}$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), 165–191.
  • E. Bedford and S. Pinchuk, Domains in $\mathbb{C}^2$ with noncompact automorphism group, Indiana Univ. Math. J. 47 (1998), no. 1, 199–222.
  • D. Cvijović, Polypseudologarithms revisited, Phys. A. 389 (2010), no. 8, 1594–1600.
  • H. P. Boas, S. Q. Fu and E. J. Straube, The Bergman kernel function: Explicit formulas and zeroes, Proc. Amer. Math. Soc. 127 (1999), no. 3, 805–811.
  • A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), no. 4, 364–374.
  • G. Francsics and N. Hanges, The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. Func. Anal. 142 (1996), no. 2, 494–510.
  • M. Engliš, Singular Berezin transforms, Compl. Anal. Oper. Theory 1 (2007), 533–548.
  • J. A. Gifford, A. V. Isaev and S. G. Krantz, On the dimensions of the automorphism groups of hyperbolic Reinhardt domains, Illinois J. Math. 44 (2000), 602–618.
  • S. G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys. 19 (1964), 1–89.
  • L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, 1963.
  • A. V. Isaev and S. G. Krantz, Hyperbolic Reinhardt domains in ${\bf C}^2$ with noncompact automorphism group, Pacific J. Math. 184 (1998), 149–160.
  • A. V. Isaev and S. G. Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999), no. 1, 1–38.
  • H. Kim, The automorphism group of an unbounded domain related to Wermer type sets, J. Math. Anal. Appl. 421 (2015), 1196–1206.
  • K.-T. Kim and S. G. Krantz, The automorphism groups of domains, Amer. Math. Monthly 112 (2005), 585–601.
  • H. Kim, V. T. Ninh and A. Yamamori, The automorphism group of a certain unbounded non-hyperbolic domain, J. Math. Anal. Appl. 409 (2014), 637–642.
  • A. Kodama, S. Krantz and D. Ma, A characterization of generalized complex ellipsoids in $C^n$ and related results, Indiana Univ. Math. J. 41 (1992), 173–195.
  • A. Kodama, On the holomorphic automorphism group of a generalized complex ellipsoid, Complex Var. Elliptic Equ. 59 (2014), no. 9, 1342–1349.
  • Ł. Kosiński, Serre problem for unbounded pseudoconvex Reinhardt domains in $\mathbb{C}^2$, J. Geom. Anal. 21 (2011), no. 4, 902–919.
  • S. G. Krantz, The automorphism groups of domains in complex space: a survey, Quaest. Math. 36 (2013), no. 2, 225–251.
  • E. Ligocka, Forelli-Rudin constructions and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257–272.
  • K. Oeljeklaus, P. Pflug and E. H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 915–928.
  • J.-D. Park, New formulas of the Bergman kernels for complex ellipsoids in $\mathbb{C}^2$, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4211–4221.
  • J.-P. Rosay, Sur une caractérisation de la boule parmi les domaines de ${\bf C}^n$ par son groupe d' automorphismes, Ann. Inst. Fourier (Grenoble) 29 (1979), 91–97.
  • S. Shimizu, Automorphisms and equivalence of bounded Reinhardt domains not containing the origin, Tohoku Math. J. 40 (1988), 119–152.
  • S. Shimizu, Automorphisms of bounded Reinhardt domains, Japan. J. Math. 15 (1989), 385–414.
  • T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Math. Ann. 235 (1978), 111–128.
  • J. Wiegerinck, Domains with finite dimensional Bergman space, Math. Z. 187 (1984), 559–562.
  • B. Wong, Characterization of the unit ball in ${\bf C}^n$ by its automorphism group, Invent. Math. 41 (1977), 253–257.
  • B. Wong, On complex manifolds with noncompact automorphism group, Contemp. Math. 332 (2003), 287–304.
  • A. Yamamori, The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function, Complex Var. Elliptic Equ. 58 (2013), no. 6, 783–793.
  • A. Yamamori, A generalization of the Forelli-Rudin construction and deflation identities, Proc. Amer. Math. Soc. 143 (2015), 1569–1581.