Properties of squeezing functions and global transformations of bounded domains
HTML articles powered by AMS MathViewer
- by Fusheng Deng, Qi’an Guan and Liyou Zhang PDF
- Trans. Amer. Math. Soc. 368 (2016), 2679-2696 Request permission
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.References
- 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, DOI 10.1016/j.jfa.2012.01.021
- Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
- Lipman Bers, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1083–1100. MR 463433, DOI 10.1090/S0002-9904-1977-14390-5
- B. Chen, Equivalence of the Bergman and Teichmüller metrics on Teichmüller spaces, eprint, arXiv:0403130.
- 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, DOI 10.2140/pjm.2012.257.319
- 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, DOI 10.1007/s12220-013-9410-0
- Miroslav Engliš, Zeroes of the Bergman kernel of Hartogs domains, Comment. Math. Univ. Carolin. 41 (2000), no. 1, 199–202. MR 1756941
- Buma L. Fridman and Daowei Ma, On exhaustion of domains, Indiana Univ. Math. J. 44 (1995), no. 2, 385–395. MR 1355403, DOI 10.1512/iumj.1995.44.1992
- Klaus Fritzsche and Hans Grauert, From holomorphic functions to complex manifolds, Graduate Texts in Mathematics, vol. 213, Springer-Verlag, New York, 2002. MR 1893803, DOI 10.1007/978-1-4684-9273-6
- 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 372252, DOI 10.1090/S0002-9947-1975-0372252-8
- Robert E. Greene, Kang-Tae Kim, and Steven G. Krantz, The geometry of complex domains, Progress in Mathematics, vol. 291, Birkhäuser Boston, Ltd., Boston, MA, 2011. MR 2799296, DOI 10.1007/978-0-8176-4622-6
- Phillip A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. (2) 94 (1971), 21–51. MR 310284, DOI 10.2307/1970733
- 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, DOI 10.1515/9783110870312
- 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
- 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
- N. G. Lloyd, Remarks on generalising Rouché’s theorem, J. London Math. Soc. (2) 20 (1979), no. 2, 259–272. MR 551453, DOI 10.1112/jlms/s2-20.2.259
- Andrea Loi and Michela Zedda, Kähler-Einstein submanifolds of the infinite dimensional projective space, Math. Ann. 350 (2011), no. 1, 145–154. MR 2785765, DOI 10.1007/s00208-010-0554-y
- 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
- Eric Overholser, Equivalence of intrinsic measures on Teichmüller space, Pacific J. Math. 235 (2008), no. 2, 297–301. MR 2386225, DOI 10.2140/pjm.2008.235.297
- Guy Roos, Weighted Bergman kernels and virtual Bergman kernels, Sci. China Ser. A 48 (2005), no. suppl., 225–237. MR 2156503, DOI 10.1007/BF02884708
- 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, DOI 10.4310/AJM.2004.v8.n1.a5
- B. Wong, Characterization of the unit ball in $\textbf {C}^{n}$ by its automorphism group, Invent. Math. 41 (1977), no. 3, 253–257. MR 492401, DOI 10.1007/BF01403050
- 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, DOI 10.1016/j.crma.2012.01.005
- Sai-Kee Yeung, Quasi-isometry of metrics on Teichmüller spaces, Int. Math. Res. Not. 4 (2005), 239–255. MR 2128436, DOI 10.1155/IMRN.2005.239
- Sai-Kee Yeung, Geometry of domains with the uniform squeezing property, Adv. Math. 221 (2009), no. 2, 547–569. MR 2508930, DOI 10.1016/j.aim.2009.01.002
- 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, DOI 10.1007/BF02903843
- W. Yin, The summarizations on research of Hua domains, Adv. Math. (China), Vol. 36, No. 2, (2007) 129-152 (in Chinese).
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
- 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).
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2679-2696
- MSC (2010): Primary 32H02, 32F45
- DOI: https://doi.org/10.1090/tran/6403
- MathSciNet review: 3449253