On Curvature Estimates of Bounded Domains

  • Liyou ZhangEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 144)


We consider the Bergman curvatures estimate for bounded domains in terms of the squeezing function. As applications, we give the asymptotic boundary behaviors of the curvatures near strictly pseudoconvex boundary points, using a recent result given by Fornaess and Wold.


Bergman curvature Squeezing function Intrinsic derivative 



The author is grateful to the organizers of the 10\(^{\text {th}}\) Korean Conference on Several Complex Variables, especially Prof. K.-T. Kim and Prof. N. Shcherbina, for their kind invitation. He would also like to thank Prof. Q.-K. Lu for many invaluable communications on this topic. Project partially supported by NSFC (No. 11371025, 11371257).


  1. [Ber]
    Bergman, S.: The kernel function and conformal mapping. American Mathematical Society, Providence, Rhode Island (1970)zbMATHGoogle Scholar
  2. [ChY]
    Cheng, S.-Y., Yau, S.-T.: On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Feffermans equation. Comm. Pure Appl. Math. 33, 507–544 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [DiF]
    Diederich, K., Fornss, J.E.: Comparison of the Bergman and the Kobayashi metric. Math. Ann. 254, 257–262 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [DFW]
    Diederich, K., Fornaess, J.E., Wold, E.F.: Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type, J. Geom. Anal. 24, 2124–2134. doi: 10.1007/s12220-013-9410-0
  5. [DiH]
    Diederich, K., Herbort, G.: Pseudoconvex domains of semiregular type. In: Contributions to Complex Analysis and Analytic Geometry, Aspects of Mathematics, vol. E26, pp. 127–161 (1994)Google Scholar
  6. [DGZ1]
    Deng, F., Guan, Q., Zhang, L.: On some properties of squeezing functions of bounded domains. Pac. J. Math. 257(2), 319–342 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [DGZ2]
    Deng, F., Guan, Q., Zhang, L.: Properties of squeezing functions and global transformations of bounded domains. arXiv:1302.5307 [math.CV] (Trans. AMS)
  8. [FoW]
    Fornaess, J.E., Wold, E.F.: An estimate for the squeezing function and estimates of invariant metrics. In: Proceedings Volume of The KSCV10. arxiv:1411.3846v1 [math.CV]
  9. [Fuk]
    Fuks, B.A.: Über geodätische Manifaltigkeiten einer invariant Geometrie. Mat. Sb. 2, 369–394 (1937)Google Scholar
  10. [GK1]
    Green, R., Krantz, S.: The stability of the Bergman kernel and the the geometry of the Bergman kernel. Bull. AMS 4, 111–115 (1981)CrossRefGoogle Scholar
  11. [GK2]
    Green, R., Krantz, S.: Deformation of complex structures, estimates for \(\bar{\partial }-\) equation, stability of the Bergman kernel. Adv. Math. 43, 1–86 (1983)CrossRefGoogle Scholar
  12. [GKK]
    Green, R., Kim, K.-T., Krantz, S.: The geometry of complex domains. Birkhauser, Boston (2011)Google Scholar
  13. [Hua]
    Hua, L.-K.: The estimation of the Riemann curvature in several complex variables. Acta Math. Sin. 4, 143–170 (1954). in ChinesezbMATHGoogle Scholar
  14. [JaP]
    Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis. De Gruyter Expositions in Mathematics, vol. 9 (1993)Google Scholar
  15. [JoS]
    Joo, J.-C., Seo, A.: Higher order asymptotic behavior of certain Kähler metrics and uniformization for strongly pseudoconvex domains. J. Korea Math. Soc. 52, 1–21 (2015)Google Scholar
  16. [KiY]
    Kim, K.-T., Yu, J.: Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains. Pac. J. Math. 176(1), 141–163 (1996)Google Scholar
  17. [KrY]
    Krantz, S., Yu, J.: On the Bergman invariant and curvatures of the Bergman metric. Ill. J. Math. 40(2), 226-244 (1996)Google Scholar
  18. [KiZ]
    Kim, K.-T., Zhang, L.: On the uniform squeezing property and the squeezing function. arXiv:1306.2390 [math.CV]
  19. [Kle]
    Klembeck, P.: Kähler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets. Indiana Univ. Math. J. 27(2), 275–282 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Kob]
    Koboyashi, S.: Geometry of bounded domains. Trans. Amer. Math. Soc. 93, 267–290 (1959)CrossRefGoogle Scholar
  21. [Kub]
    Kubota, Y.: A note on holomorphic imbeddings of the classical Cartan domains into the unit ball. Proc. Amer. Math. Soc. 85(1), 65–68 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [LSY1]
    Liu, K.-F., Sun, X.-F., Yau, S.-T.: Canonical metrics on the moduli space of Riemann surfaces. I. J. Differ. Geom. 68(3), 571–637 (2004)MathSciNetzbMATHGoogle Scholar
  23. [Lu1]
    Lu, Q.-K.: On Kähler manifolds with constant curvature. Acta Math. Sin. 16, 269–281 (1966)zbMATHGoogle Scholar
  24. [Lu2]
    Lu, Q.-K.: The estimation of the intrinsic derivatives of the analytic mapping of bounded domains. Sci. Sin. Spec. Ser. II, 1–17 (1979)Google Scholar
  25. [Lu3]
    Lu, Q.-K.: Holomorphic invariant forms of a bounded domain. Sci. China Ser. A 51, 1945–1964 (2008)Google Scholar
  26. [Lu4]
    Lu, Q.-K.: On the lower bounds of the curvatures in a bounded domain. Sci. China Ser. A 58, 1–10 (2015)Google Scholar
  27. [NS1]
    Nemirovskii, S., Shafikov, R.: Uniformization of strictly pseudoconvex domains. I. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 69(6), 115–130 (2005) (translation in Izv. Math. 69(6), 1189–1202 (2005))Google Scholar
  28. [NS2]
    Nemirovskii, S., Shafikov, R.: Uniformization of strictly pseudoconvex domains. II. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 69(6), 131–138 (2005) (translation in Izv. Math. 69(6), 1203–1210 (2005))Google Scholar
  29. [Noz]
    Nozarjan, E.: Estimates of Ricci curvature. Nauk. Arm. SSR Ser. Mat. 8, 418–423 (1973)Google Scholar
  30. [Yeu]
    Yeung, S.-K.: Geometry of domains with the uniform squeezing property. Adv. Math. 221(2), 547–569 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.School of Mathematical ScienceCapital Normal UniversityBeijingPeople’s Republic of China

Personalised recommendations