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Properties of squeezing functions and global transformations of bounded domains


Authors: Fusheng Deng, Qi’an Guan and Liyou Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 2679-2696
MSC (2010): Primary 32H02, 32F45
DOI: https://doi.org/10.1090/tran/6403
Published electronically: August 19, 2015
MathSciNet review: 3449253
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Abstract: The central purpose of the present paper is to study boundary behaviors of squeezing functions on some bounded domains. We prove that the squeezing function of any strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any bounded domain near its globally strongly convex boundary points. We also study the stability properties of squeezing functions on a sequence of bounded domains, and give some comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new proofs of several well-known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems for further study are proposed.


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  • [1] Heungju Ahn and Jong-Do Park, The explicit forms and zeros of the Bergman kernel function for Hartogs type domains, J. Funct. Anal. 262 (2012), no. 8, 3518-3547. MR 2889166, https://doi.org/10.1016/j.jfa.2012.01.021
  • [2] Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), no. 2, 94-97. MR 0111834 (22:2694)
  • [3] Lipman Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1083-1100. MR 0463433 (57:3384)
  • [4] B. Chen, Equivalence of the Bergman and Teichmüller metrics on Teichmüller spaces, eprint, arXiv:0403130.
  • [5] Fusheng Deng, Qian Guan, and Liyou Zhang, Some properties of squeezing functions on bounded domains, Pacific J. Math. 257 (2012), no. 2, 319-341. MR 2972468, https://doi.org/10.2140/pjm.2012.257.319
  • [6] K. Diederich, J. E. Fornæss, and E. F. Wold, Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite $ 1$-type, J. Geom. Anal. 24 (2014), no. 4, 2124-2134. MR 3261733, https://doi.org/10.1007/s12220-013-9410-0
  • [7] Miroslav Engliš, Zeroes of the Bergman kernel of Hartogs domains, Comment. Math. Univ. Carolin. 41 (2000), no. 1, 199-202. MR 1756941 (2001f:32002)
  • [8] Buma L. Fridman and Daowei Ma, On exhaustion of domains, Indiana Univ. Math. J. 44 (1995), no. 2, 385-395. MR 1355403 (96h:32028), https://doi.org/10.1512/iumj.1995.44.1992
  • [9] Klaus Fritzsche and Hans Grauert, From holomorphic functions to complex manifolds, Graduate Texts in Mathematics, vol. 213, Springer-Verlag, New York, 2002. MR 1893803 (2003g:32001)
  • [10] Ian Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $ C^{n}$ with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219-240. MR 0372252 (51 #8468)
  • [11] Robert E. Greene, Kang-Tae Kim, and Steven G. Krantz, The geometry of complex domains, Progress in Mathematics, vol. 291, Birkhäuser Boston, Inc., Boston, MA, 2011. MR 2799296 (2012c:32001)
  • [12] Phillip A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. (2) 94 (1971), 21-51. MR 0310284 (46 #9385)
  • [13] Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, de Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120 (94k:32039)
  • [14] Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Canonical metrics on the moduli space of Riemann surfaces. I, J. Differential Geom. 68 (2004), no. 3, 571-637. MR 2144543 (2007g:32009)
  • [15] Kefeng Liu, Xiaofeng Sun, and Shing-Tung Yau, Canonical metrics on the moduli space of Riemann surfaces. II, J. Differential Geom. 69 (2005), no. 1, 163-216. MR 2169586 (2007g:32010)
  • [16] N. G. Lloyd, Remarks on generalising Rouché's theorem, J. London Math. Soc. (2) 20 (1979), no. 2, 259-272. MR 551453 (80m:32029), https://doi.org/10.1112/jlms/s2-20.2.259
  • [17] Andrea Loi and Michela Zedda, Kähler-Einstein submanifolds of the infinite dimensional projective space, Math. Ann. 350 (2011), no. 1, 145-154. MR 2785765 (2012f:32031), https://doi.org/10.1007/s00208-010-0554-y
  • [18] Ngaiming Mok and Shing-Tung Yau, Completeness of the Kähler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, The mathematical heritage of Henri Poincaré, Part 1 (Bloomington, Ind., 1980) Proc. Sympos. Pure Math., vol. 39, Amer. Math. Soc., Providence, RI, 1983, pp. 41-59. MR 720056 (85j:53068)
  • [19] Eric Overholser, Equivalence of intrinsic measures on Teichmüller space, Pacific J. Math. 235 (2008), no. 2, 297-301. MR 2386225 (2009h:32022), https://doi.org/10.2140/pjm.2008.235.297
  • [20] Guy Roos, Weighted Bergman kernels and virtual Bergman kernels, Sci. China Ser. A 48 (2005), no. suppl., 225-237. MR 2156503 (2006e:32007), https://doi.org/10.1007/BF02884708
  • [21] An Wang, Weiping Yin, Liyou Zhang, and Wenjuan Zhang, The Einstein-Kähler metric with explicit formulas on some non-homogeneous domains, Asian J. Math. 8 (2004), no. 1, 39-49. MR 2128296 (2005k:32026), https://doi.org/10.4310/AJM.2004.v8.n1.a5
  • [22] B. Wong, Characterization of the unit ball in $ {\bf C}^{n}$ by its automorphism group, Invent. Math. 41 (1977), no. 3, 253-257. MR 0492401 (58 #11521)
  • [23] Atsushi Yamamori, A remark on the Bergman kernels of the Cartan-Hartogs domains, C. R. Math. Acad. Sci. Paris 350 (2012), no. 3-4, 157-160 (English, with English and French summaries). MR 2891103, https://doi.org/10.1016/j.crma.2012.01.005
  • [24] Sai-Kee Yeung, Quasi-isometry of metrics on Teichmüller spaces, Int. Math. Res. Not. 4 (2005), 239-255. MR 2128436 (2005m:32028), https://doi.org/10.1155/IMRN.2005.239
  • [25] Sai-Kee Yeung, Geometry of domains with the uniform squeezing property, Adv. Math. 221 (2009), no. 2, 547-569. MR 2508930 (2010b:32034), https://doi.org/10.1016/j.aim.2009.01.002
  • [26] Weiping Yin, The Bergman kernels on super-Cartan domains of the first type, Sci. China Ser. A 43 (2000), no. 1, 13-21. MR 1766243 (2001e:32004), https://doi.org/10.1007/BF02903843
  • [27] W. Yin, The summarizations on research of Hua domains, Adv. Math. (China), Vol. 36, No. 2, (2007) 129-152 (in Chinese).

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Additional Information

Fusheng Deng
Affiliation: School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
Email: fshdeng@ucas.ac.cn

Qi’an Guan
Affiliation: Beijing International Center for Mathematical Research, and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
Email: guanqian@math.pku.edu.cn

Liyou Zhang
Affiliation: School of Mathematical Sciences, Capital Normal University, Beijing 100048, People’s Republic of China
Email: zhangly@cnu.edu.cn

DOI: https://doi.org/10.1090/tran/6403
Keywords: Squeezing function, homogeneous regular domain, globally strongly convex boundary point
Received by editor(s): April 24, 2013
Received by editor(s) in revised form: January 26, 2014
Published electronically: August 19, 2015
Additional Notes: The authors were partially supported by NSFC grants and BNSF(No.1122010).
Article copyright: © Copyright 2015 American Mathematical Society

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