Illinois Journal of Mathematics

Boundary behavior of the Carathéodory and Kobayashi–Eisenman volume elements

Diganta Borah and Debaprasanna Kar

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


We study the boundary asymptotics of the Carathéodory and Kobayashi–Eisenman volume elements on smoothly bounded convex finite type domains and Levi corank one domains.

Article information

Illinois J. Math., Volume 64, Number 2 (2020), 151-168.

Received: 1 April 2019
Revised: 12 December 2019
First available in Project Euclid: 1 May 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32F45: Invariant metrics and pseudodistances
Secondary: 32T27: Geometric and analytic invariants on weakly pseudoconvex boundaries 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)


Borah, Diganta; Kar, Debaprasanna. Boundary behavior of the Carathéodory and Kobayashi–Eisenman volume elements. Illinois J. Math. 64 (2020), no. 2, 151--168. doi:10.1215/00192082-8303461.

Export citation


  • [1] T. J. Barth, Convex domains and Kobayashi hyperbolicity, Proc. Amer. Math. Soc. 79 (1980), no. 4, 556–558.
  • [2] E. Bedford and S. Pinchuk, Domains in $\mathbf{C}^{n+1}$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165–191.
  • [3] F. Berteloot and G. Cœuré, Domaines de $\mathbf{C}^{2}$, pseudoconvexes et de type fini ayant un groupe non compact d’automorphismes, Ann. Inst. Fourier (Grenoble) 41 (1991), no. 1, 77–86.
  • [4] W. S. Cheung and B. Wong, An integral inequality of an intrinsic measure on bounded domains in $\mathbf{C}^{n}$, Rocky Mountain J. Math. 22 (1992), no. 3, 825–836.
  • [5] S. Cho, Boundary behavior of the Bergman kernel function on some pseudoconvex domains in $\mathbf{C}^{n}$, Trans. Amer. Math. Soc. 345 (1994), no. 2, 803–817.
  • [6] I. M. Dektyarev, Criterion for the equivalence of hyperbolic manifolds, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 73–74 (Russian).
  • [7] H. Gaussier, Characterization of convex domains with noncompact automorphism group, Michigan Math. J. 44 (1997), no. 2, 375–388.
  • [8] H. Gaussier, Tautness and complete hyperbolicity of domains in $\mathbf{C}^{n}$, Proc. Amer. Math. Soc. 127 (1999), no. 1, 105–116.
  • [9] I. Graham and H. Wu, Characterizations of the unit ball $B^{n}$ in complex Euclidean space, Math. Z. 189 (1985), no. 4, 449–456.
  • [10] R. E. Greene and S. G. Krantz, “Characterizations of certain weakly pseudoconvex domains with noncompact automorphism groups” in Complex Analysis, Lect. Notes Math. 1268, Springer, Berlin, 1987, 121–157.
  • [11] R. E. Greene and S. G. Krantz, “Biholomorphic self-maps of domains” in Complex Analysis II, Lect. Notes Math. 1276, Springer, Berlin, 1987, 136–207.
  • [12] S. G. Krantz, The Kobayashi metric, extremal discs, and biholomorphic mappings, Complex Var. Elliptic Equ. 57 (2012), no. 1, 1–14.
  • [13] D. Ma, Boundary behavior of invariant metrics and volume forms on strongly pseudoconvex domains, Duke Math. J. 63 (1991), no. 3, 673–697.
  • [14] P. Mahajan and K. Verma, A comparison of two biholomorphic invariants, Internat. J. Math. 30 (2019), no. 1, 1950012, 16 pp.
  • [15] J. D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109 (1994), no. 1, 108–139.
  • [16] N. Nikolov, Behavior of the squeezing function near h-extendible boundary points, Proc. Amer. Math. Soc. 146 (2018), no. 8, 3455–3457.
  • [17] N. Nikolov and P. J. Thomas, Comparison of the Bergman kernel and the Carathéodory–Eisenman volume, Proc. Amer. Math. Soc. 147 (2019), no. 11, 4915-4919.
  • [18] N. Nikolov and K. Verma, On the squeezing function and Fridman invariants, to appear in J. Geom. Anal.
  • [19] J.-P. Rosay, Sur une caractérisation de la boule parmi les domaines de $\mathbf{C}^{n}$ par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, ix, 91–97.
  • [20] D. D. Thai and N. V. Thu, Characterization of domains in $\mathbb{C}^{n}$ by their noncompact automorphism groups, Nagoya Math. J. 196 (2009), 135–160.
  • [21] B. Wong, Characterization of the unit ball in $\mathbf{C}^{n}$ by its automorphism group, Invent. Math. 41 (1977), no. 3, 253–257.
  • [22] J. Y. Yu, Weighted boundary limits of the generalized Kobayashi–Royden metrics on weakly pseudoconvex domains, Trans. Amer. Math. Soc. 347 (1995), no. 2, 587–614.
  • [23] A. M. Zimmer, Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Math. Ann. 365 (2016), nos. 3–4, 1425–1498.
  • [24] A. Zimmer, A gap theorem for the complex geometry of convex domains, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7489–7509.
  • [25] A. Zimmer, Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents, Math. Ann. 374 (2019), nos. 3–4, 1811–1844.