Abstract
This survey paper consists of two folds. First of all, we recall the concept of intrinsic derivative which was introduced by Lu (1979) and the related works due to Lu in his last ten years, including the holomorphically isometric embedding into the infinite dimensional Grassmann manifold and the Bergman curvature estimates for bounded domains in ℂn. Inspired by Lu’s idea, we give the lower and upper bounds estimates for the Bergman curvatures in terms of the squeezing function—one concept originally introduced by Deng et al. (2012). Finally, we survey some recent progress on the asymptotic behaviors for Bergman curvatures near the strictly pseudoconvex boundary points and present some open problems on the squeezing functions of bounded domains in ℂn.
Similar content being viewed by others
References
Cheng S-Y, Yau S-T. On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation. Comm Pure Appl Math, 1980, 33: 507–544
Cho S, Song T. The holomorphic sectional curvature of the Bergman metric in ℂn. Houston J Math, 1995, 21: 441–448
Deng F, Fornaess J E, Wold E F. Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces. ArXiv:1607.02755, 2016
Deng F, Guan Q, Zhang L. Some properties of squeezing functions on bounded domains. Pacific J Math, 2012, 257: 319–341
Deng F, Guan Q, Zhang L. Properties of squeezing functions and global transformations of bounded domains. Trans Amer Math Soc, 2016, 368: 2679–2696
Diederich K, Fornaess J E. Comparison of the Bergman and the Kobayashi metric. Math Ann, 1980, 254: 257–262
Diederich K, Fornaess J E. Boundary behavior of the Bergman metric. ArXiv:1504.02950, 2015
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, 2014, 24: 2124–2134
Diederich K, Fornaess J E, Wold E F. A characterization of the ball. ArXiv:1604.05057, 2016
Diederich K, Herbort G. Pseudoconvex domains of semiregular type. In: Contributions to Complex Analysis and Analytic Geometry. Analyse Complexe et Géométrie Analytique. Aspects of Mathematics. vol. E26. Fachmedien- Wiesbaden: Springer, 1994, 127–161
Dinew Ż. An example for the holomorphic sectional curvature of the Bergman metric. Ann Polon Math, 2010, 98: 147–167
Dinew Ż. On the Bergman representative coordinates. Sci China Math, 2011, 54: 1357–1374
Engliš M. Boundary behaviour of the Bergman invariant and related quantities. Monatsh Math, 2008, 154: 19–37
Fornaess J E, Rong F. Estimate of the squeezing function for a class of bounded domains. ArXiv:1606.01335, 2016
Fornaess J E, Shcherbina N. A domain with non-plurisubharmonic squeezing function. ArXiv:1604.01480v2, 2016
Fornaess J E, Wold E F. An estimate for the squeezing function and estimates of invariant metrics. In: Complex Analysis and Geometry. Springer Proceedings in Mathematics Statistics, vol. 144. Tokyo: Springer, 2015, 135–147
Fornaess J E, Wold E F. A non-strictly pseudoconvex domain for which the squeezing function tends to one towards the boundary. ArXiv:1611.04464v1, 2016
Fu S. Geometry of Reinhardt domains of finite type in ℂ2. J Geom Anal, 1996, 6: 407–431
Greene R, Kim K-T, Krantz S. The Geometry of Complex Domains. Boston: Birkhäuser, 2011
Greene R, Krantz S. The stability of the Bergman kernel and the the geometry of the Bergman kernel. Bull Amer Math Soc, 1981, 4: 111–115
Greene R, Krantz S. Deformation of complex structures, estimates for ∂̄-equation, stability of the Bergman kernel. Adv Math, 1983, 43: 1–86
Herbort G. An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded. Ann Polon Math, 2007, 92: 29–39
Joo J-C, Seo A. Higher order asymptotic behavior of certain Kähler metrics and uniformization for strongly pseudoconvex domains. J Korean Math Soc, 2015, 52: 113–124
Joo S, Kim K-T. On the boundary points at which the squeezing function tends to one. ArXiv:1611.08356v2, 2016
Kim K-T, Yu J. Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains. Pacific J Math, 1996, 176: 141–163
Kim K-T, Zhang L. On the uniform squeezing property of bounded convex domains in ℂn. Pacific J Math, 2016, 282: 341–358
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, 1978, 27: 275–282
Krantz S, Yu J. On the Bergman invariant and curvatures of the Bergman metric. Illinois J Math, 1996, 40: 226–244
Kubota Y. A note on holomorphic imbeddings of the classical Cartan domains into the unit ball. Proc Amer Math Soc, 1982, 85: 65–68
Liu K, Sun X, Yau S-T. Canonical metrics on the moduli space of Riemann surfaces, I. J Differential Geom, 2004, 68: 571–637
Lu Q-K. The estimation of the intrinsic derivatives of the analytic mapping of bounded domains. Sci Sinica, 1979, S1: 1–17
Lu Q-K. Holomorphic invariant forms of a bounded domain. Sci China Ser A, 2008, 51: 1945–1964
Lu Q-K. On the lower bounds of the curvatures in a bounded domain. Sci China Math, 2015, 58: 1–10
McNeal J D. Holomorphic sectional curvature of some pseudoconvex domains. Proc Amer Math Soc, 1989, 107: 113–117
Nemirovskii S, Shafikov R. Uniformization of strictly pseudoconvex domains. I (in Russian). Izv Ross Akad Nauk Ser Mat, 2005, 69: 115–130; translation in Izv Math, 2005, 69: 1189–1202
Nemirovskii S, Shafikov R. Uniformization of strictly pseudoconvex domains. II (in Russian). Izv Ross Akad Nauk Ser Mat, 2005, 69: 131–138; translation in Izv Math, 2005, 69: 1203–1210
Wold E F. Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk. ArXiv:1606.01794, 2016
Yeung S-K. Geometry of domains with the uniform squeezing property. Adv Math, 2009, 221: 547–569
Zhang L. On curvature estimates of bounded domains. In: Complex Analysis and Geometry. Springer Proceedings in Mathematics Statistics, vol. 144. Tokyo: Springer, 2015, 353–367
Zimmer A. A gap theorem for the complex geometry of convex domains. ArXiv:1609.07050, 2016
Zwonek W. Asymptotic behavior of the sectional curvature of the Bergman metric for annuli. Ann Polon Math, 2010, 98: 291–299
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11371025 and 11671270). The author dedicates the paper for the memory of the late Professor Qikeng Lu, from whom the author has learnt not only mathematics, but also the attitude towards life and truth. The author also thanks the referees’ comments which made the paper more readable.
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of Professor LU QiKeng (1927–2015)
Rights and permissions
About this article
Cite this article
Zhang, L. Intrinsic derivative, curvature estimates and squeezing function. Sci. China Math. 60, 1149–1162 (2017). https://doi.org/10.1007/s11425-016-9043-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9043-8