The space of convex domains in complex Euclidean space
In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kähler–Einstein metric are all con...
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| Vydáno v: | The Journal of geometric analysis Ročník 30; číslo 2; s. 1312 - 1358 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.04.2020
Springer Nature B.V Springer |
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| ISSN: | 1050-6926, 1559-002X |
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| Abstract | In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kähler–Einstein metric are all continuous on the space of convex domains. The group of affine automorphisms acts on this space and we also describe the orbit closures for some special classes of domains. |
|---|---|
| AbstractList | In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kähler–Einstein metric are all continuous on the space of convex domains. The group of affine automorphisms acts on this space and we also describe the orbit closures for some special classes of domains. |
| Author | Gaussier, Hervé Zimmer, Andrew |
| Author_xml | – sequence: 1 givenname: Hervé surname: Gaussier fullname: Gaussier, Hervé email: herve.gaussier@univ-grenoble-alpes.fr organization: Univ. Grenoble Alpes, CNRS, IF – sequence: 2 givenname: Andrew surname: Zimmer fullname: Zimmer, Andrew organization: Department of Mathematics, Louisiana State University |
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| Cites_doi | 10.1006/jfan.1998.3317 10.1002/cpa.3160330404 10.1090/tran/6403 10.1016/0022-1236(92)90029-I 10.1090/pspum/039.1/720056 10.1090/pspum/052.1/1128522 10.1016/j.aim.2009.01.002 10.1215/S0012-7094-76-04367-2 10.2140/pjm.2016.282.341 10.1515/FORUM.2009.039 10.1007/s00208-017-1546-y 10.1007/BF02392734 10.1007/BF02201775 10.1090/conm/101/1034986 10.2140/pjm.1996.176.141 10.1515/9783110253863 10.1090/pspum/052.2/1128543 10.1007/b97238 10.1307/mmj/1029005712 10.1007/s12220-013-9410-0 10.1090/S0002-9939-1980-0572300-3 10.4310/jdg/1116508767 10.5802/aif.768 10.1007/978-0-8176-4622-6 10.1006/aima.1998.1821 10.4310/jdg/1121540343 10.1007/978-3-319-65837-7 10.1007/BF02921382 10.1006/aima.1994.1082 10.1090/pspum/052.2/1128547 10.1007/s00208-015-1278-9 10.1142/S0129167X17500318 10.1007/BF01403050 |
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| Keywords | Invariant metrics and pseudodistances Convex domains 32F17 Primary 32F45 |
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| References | Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean Press, Rende (1989) Pinchuk, S.: The scaling method and holomorphic mappings. In: Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 151–161. American Mathematical Society, Providence (1991) FuSStraubeEJCompactness of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problem on convex domainsJ. Funct. Anal.19981592629641 HörmanderLAn Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library19903AmsterdamNorth-Holland Publishing Co. NikolovNAndreevLBoundary behavior of the squeezing functions of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}-convex domains and plane domainsInt. J. Math.20172851750031,5 KimK-TZhangLOn the uniform squeezing property of bounded convex domains in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{C}^n$$\end{document}Pacific J. Math.20162822341358 GreeneREKimK-TKrantzSGThe Geometry of Complex Domains. Progress in Mathematics2011BostonBirkhäuser Boston Inc. BarthTJConvex domains and Kobayashi hyperbolicityProc. Am. Math. Soc.1980794556558 DiederichKFornæssJEWoldEFExposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-typeJ. Geom. Anal.201424421242134 Graham, I.: Sharp constants for the Koebe theorem and for estimates of intrinsic metrics on convex domains. In: Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 233–238. American Mathematical Society, Providence (1991) LiuKSunXYauS-TCanonical metrics on the moduli space of Riemann surfaces. IIJ. Differ. Geom.2005691163216 LiuKSunXYauS-TCanonical metrics on the moduli space of Riemann surfaces. IJ. Differ. Geom.2004683571637 RosayJ-PSur une caractérisation de la boule parmi les domaines de Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}^n$$\end{document} par son groupe d’automorphismesAnn. Inst. Fourier19792949197 WongBCharacterization of the unit ball in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}^n$$\end{document} by its automorphism groupInvent. Math.197741253257 Mok, N., Yau, S.-T.: Completeness of the Kähler–Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions. The mathematical heritage of Henri Poincare, Proceedings of Symposium Pure Mathematics 39, Part 1, Bloomington/Indiana 1980, pp. 41–59 (1983) Frankel, S.: Applications of affine geometry to geometric function theory in several complex variables. I. Convergent rescalings and intrinsic quasi-isometric structure. In: Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 183–208. American Mathematical Society, Providence (1991) KimK-TJiyeYuBoundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domainsPacific J. Math.19961761141163 FornæssJERongFEstimate of the squeezing function for a class of bounded domainsMath. Ann.20183713–410871094 AxlerSBourdonPRameyWHarmonic Function Theory. Graduate Texts in Mathematics1992New YorkSpringer BedfordEPinchukSIConvex domains with noncompact groups of automorphismsMat. Sb.19941855326 FrankelSComplex geometry of convex domains that cover varietiesActa Math.19891631–2109149 BracciFSaraccoAHyperbolicity in unbounded convex domainsForum Math.2009215815825 IsaevAVKrantzSGDomains with non-compact automorphism group: a surveyAdv. Math.19991461138 McNealJDEstimates on the Bergman kernels of convex domainsAdv. Math.19941091108139 Bracci, F., Gaussier, H., Zimmer, A.: The geometry of domains with negatively pinched Kähler metrics. arXiv:1810.11389 (2018) LempertLHolomorphic retracts and intrinsic metrics in convex domainsAnal. Math.19828257261 Blanc-CentiLMetrical and Dynamical Aspects in Complex Analysis. Lecture Notes in Mathematics2017BerlinSpringer International Publishing RamadanovISur une propriété de la fonction de BergmanC. R. Acad. Bulg. Sci.196720759762 YeungS-KGeometry of domains with the uniform squeezing propertyAdv. Math.20092212547569 Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis, vol. 9, 2nd extended edn. Walter de Gruyter, Berlin (2013) Frankel, S.: Affine approach to complex geometry. In: Recent Developments in Geometry (Los Angeles, CA, 1987), Contemporary Mathematics, vol. 101, pp. 263–286. American Mathematical Society, Providence (1989) YangPCOn Kähler manifolds with negative holomorphic bisectional curvatureDuke Math. J.197643871874 ZimmerAMGromov hyperbolicity and the Kobayashi metric on convex domains of finite typeMath. Ann.20163653–414251498 ChengS-YYauS-TOn the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equationCommun. Pure Appl. Math.198033507544 McNealJDConvex domains of finite typeJ. Funct. Anal.19921082361373 DengFGuanQZhangLProperties of squeezing functions and global transformations of bounded domainsTrans. Am. Math. Soc.2016368426792696 GaussierHCharacterization of convex domains with noncompact automorphism groupMichigan Math. J.1997442375388 Kim, K.-T.: Geometry of bounded domains and the scaling techniques in several complex variables. Lecture Notes Series, vol. 13. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1993) Wu, D., Yau, S.-T.: Invariant metrics on negatively pinched complete Kähler manifolds. ArXiv e-prints(2017) BoasHPStraubeEJOn equality of line type and variety type of real hypersurfaces in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ C}^n$$\end{document}J. Geom. Anal.1992229598 346_CR18 K-T Kim (346_CR24) 1996; 176 JE Fornæss (346_CR12) 2018; 371 RE Greene (346_CR19) 2011 PC Yang (346_CR38) 1976; 43 K-T Kim (346_CR25) 2016; 282 I Ramadanov (346_CR34) 1967; 20 B Wong (346_CR36) 1977; 41 K Diederich (346_CR10) 2014; 24 346_CR22 JD McNeal (346_CR30) 1994; 109 346_CR23 AM Zimmer (346_CR40) 2016; 365 S Frankel (346_CR14) 1989; 163 S-K Yeung (346_CR39) 2009; 221 K Liu (346_CR28) 2005; 69 346_CR5 346_CR1 HP Boas (346_CR7) 1992; 2 F Bracci (346_CR8) 2009; 21 L Hörmander (346_CR20) 1990 L Blanc-Centi (346_CR4) 2017 K Liu (346_CR27) 2004; 68 F Deng (346_CR11) 2016; 368 L Lempert (346_CR26) 1982; 8 JD McNeal (346_CR29) 1992; 108 AV Isaev (346_CR21) 1999; 146 E Bedford (346_CR6) 1994; 185 N Nikolov (346_CR32) 2017; 28 J-P Rosay (346_CR35) 1979; 29 TJ Barth (346_CR3) 1980; 79 346_CR31 S-Y Cheng (346_CR9) 1980; 33 H Gaussier (346_CR17) 1997; 44 346_CR33 346_CR13 S Fu (346_CR16) 1998; 159 S Axler (346_CR2) 1992 346_CR15 346_CR37 |
| References_xml | – reference: BarthTJConvex domains and Kobayashi hyperbolicityProc. Am. Math. Soc.1980794556558 – reference: LiuKSunXYauS-TCanonical metrics on the moduli space of Riemann surfaces. IJ. Differ. Geom.2004683571637 – reference: ZimmerAMGromov hyperbolicity and the Kobayashi metric on convex domains of finite typeMath. Ann.20163653–414251498 – reference: Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry. Mediterranean Press, Rende (1989) – reference: ChengS-YYauS-TOn the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equationCommun. Pure Appl. Math.198033507544 – reference: LiuKSunXYauS-TCanonical metrics on the moduli space of Riemann surfaces. IIJ. Differ. Geom.2005691163216 – reference: Bracci, F., Gaussier, H., Zimmer, A.: The geometry of domains with negatively pinched Kähler metrics. arXiv:1810.11389 (2018) – reference: WongBCharacterization of the unit ball in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}^n$$\end{document} by its automorphism groupInvent. Math.197741253257 – reference: BedfordEPinchukSIConvex domains with noncompact groups of automorphismsMat. Sb.19941855326 – reference: Wu, D., Yau, S.-T.: Invariant metrics on negatively pinched complete Kähler manifolds. ArXiv e-prints(2017) – reference: FrankelSComplex geometry of convex domains that cover varietiesActa Math.19891631–2109149 – reference: BracciFSaraccoAHyperbolicity in unbounded convex domainsForum Math.2009215815825 – reference: BoasHPStraubeEJOn equality of line type and variety type of real hypersurfaces in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ C}^n$$\end{document}J. Geom. Anal.1992229598 – reference: DengFGuanQZhangLProperties of squeezing functions and global transformations of bounded domainsTrans. Am. Math. Soc.2016368426792696 – reference: NikolovNAndreevLBoundary behavior of the squeezing functions of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}-convex domains and plane domainsInt. J. Math.20172851750031,5 – reference: Kim, K.-T.: Geometry of bounded domains and the scaling techniques in several complex variables. Lecture Notes Series, vol. 13. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1993) – reference: Blanc-CentiLMetrical and Dynamical Aspects in Complex Analysis. Lecture Notes in Mathematics2017BerlinSpringer International Publishing – reference: YeungS-KGeometry of domains with the uniform squeezing propertyAdv. Math.20092212547569 – reference: GaussierHCharacterization of convex domains with noncompact automorphism groupMichigan Math. J.1997442375388 – reference: RamadanovISur une propriété de la fonction de BergmanC. R. Acad. Bulg. Sci.196720759762 – reference: Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis, vol. 9, 2nd extended edn. Walter de Gruyter, Berlin (2013) – reference: AxlerSBourdonPRameyWHarmonic Function Theory. Graduate Texts in Mathematics1992New YorkSpringer – reference: Pinchuk, S.: The scaling method and holomorphic mappings. In: Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 151–161. American Mathematical Society, Providence (1991) – reference: GreeneREKimK-TKrantzSGThe Geometry of Complex Domains. Progress in Mathematics2011BostonBirkhäuser Boston Inc. – reference: KimK-TJiyeYuBoundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domainsPacific J. Math.19961761141163 – reference: FuSStraubeEJCompactness of the ∂¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\partial }}$$\end{document}-Neumann problem on convex domainsJ. Funct. Anal.19981592629641 – reference: DiederichKFornæssJEWoldEFExposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-typeJ. Geom. Anal.201424421242134 – reference: RosayJ-PSur une caractérisation de la boule parmi les domaines de Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}^n$$\end{document} par son groupe d’automorphismesAnn. Inst. Fourier19792949197 – reference: Mok, N., Yau, S.-T.: Completeness of the Kähler–Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions. The mathematical heritage of Henri Poincare, Proceedings of Symposium Pure Mathematics 39, Part 1, Bloomington/Indiana 1980, pp. 41–59 (1983) – reference: HörmanderLAn Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library19903AmsterdamNorth-Holland Publishing Co. – reference: Frankel, S.: Affine approach to complex geometry. In: Recent Developments in Geometry (Los Angeles, CA, 1987), Contemporary Mathematics, vol. 101, pp. 263–286. American Mathematical Society, Providence (1989) – reference: IsaevAVKrantzSGDomains with non-compact automorphism group: a surveyAdv. Math.19991461138 – reference: McNealJDEstimates on the Bergman kernels of convex domainsAdv. Math.19941091108139 – reference: FornæssJERongFEstimate of the squeezing function for a class of bounded domainsMath. Ann.20183713–410871094 – reference: Graham, I.: Sharp constants for the Koebe theorem and for estimates of intrinsic metrics on convex domains. In: Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 233–238. American Mathematical Society, Providence (1991) – reference: Frankel, S.: Applications of affine geometry to geometric function theory in several complex variables. I. Convergent rescalings and intrinsic quasi-isometric structure. In: Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 183–208. American Mathematical Society, Providence (1991) – reference: KimK-TZhangLOn the uniform squeezing property of bounded convex domains in Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{C}^n$$\end{document}Pacific J. Math.20162822341358 – reference: YangPCOn Kähler manifolds with negative holomorphic bisectional curvatureDuke Math. J.197643871874 – reference: McNealJDConvex domains of finite typeJ. Funct. Anal.19921082361373 – reference: LempertLHolomorphic retracts and intrinsic metrics in convex domainsAnal. Math.19828257261 – volume: 159 start-page: 629 issue: 2 year: 1998 ident: 346_CR16 publication-title: J. Funct. 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| SubjectTerms | Abstract Harmonic Analysis an Important Tool in Several Complex Variables Automorphisms Closures Convex and Discrete Geometry Differential Geometry Domains Dynamical Systems and Ergodic Theory Euclidean geometry Euclidean space Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Invariant Metrics Mathematics Mathematics and Statistics Topology |
| Title | The space of convex domains in complex Euclidean space |
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