A survey on Bergman completeness. (English) Zbl 1326.32005
Bracci, Filippo (ed.) et al., Complex analysis and geometry. KSCV 10. Proceddings of the 10th symposium, Gyeongju, Korea, August 7–11, 2014. Tokyo: Springer (ISBN 978-4-431-55743-2/hbk; 978-4-431-55744-9/ebook). Springer Proceedings in Mathematics & Statistics 144, 99-117 (2015).
Summary: We provide a survey of results on Bergman completeness of open complex manifolds.
For the entire collection see [Zbl 1328.32001].
For the entire collection see [Zbl 1328.32001].
MSC:
| 32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |
| 32A36 | Bergman spaces of functions in several complex variables |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
References:
| [1] | Ahlfors, L., Sario, L.: Riemann Sufaces. Princeton,New Jersey, Princeton University Press(1960) · Zbl 0196.33801 |
| [2] | Ahn, T., Gaussier, H., Kim, K.-T.: Bergman and Carathéodory metrics of the Kohn-Nirenberg domains, arXiv:1406.3406 |
| [3] | Berndtsson, B., Lempert, L.: A proof of the Ohsawa-Takegoshi theorem with sharp estimates, arXiv:1407.4946v1 · Zbl 1360.32006 |
| [4] | Blocki, Z., Estimates for the complex Monge-Ampere operator, Bull. Pol. Acad. Sci., 41, 151-157 (1993) · Zbl 0795.32003 |
| [5] | Blocki, Z., The complex Monge-Ampere operator in hyperconvex domains, Annali della Scuola Norm. Sup. Pisa, 23, 721-747 (1996) · Zbl 0878.31003 |
| [6] | Blocki, Z., The Bergman metric and the pluricomplex Green function, Trans. Am. Math. Soc., 357, 2613-2625 (2004) · Zbl 1071.32008 · doi:10.1090/S0002-9947-05-03738-4 |
| [7] | Blocki, Z., Suita conjecture and the Ohsawa-Takegoshi extension theorem, Invent. Math., 193, 149-158 (2013) · Zbl 1282.32014 · doi:10.1007/s00222-012-0423-2 |
| [8] | Z. Blocki, A lower bound for the Bergman kernel and the Bourgain-Milman inequality, GAFA Seminar Notes, Lect. Notes in Math., to appear · Zbl 1321.32003 |
| [9] | Blocki, Z., Pflug, P.: Hyperconvexity and Bergman completeness. Nagoya Math. J. 151, 221-225 (1998) · Zbl 0916.32016 |
| [10] | Bremermann, HJ, Holomorphic continuation of the kernel function and the Bergman metric in several complex variables (1955), Michigan: Lectures on Functions of a Complex Variable, Michigan · Zbl 0067.30704 |
| [11] | Chen, B-Y, Completeness of the Bergman metric on non-smooth pseudoconvex domains, Ann. Polon. Math., 71, 241-251 (1999) · Zbl 0937.32014 |
| [12] | Chen, B-Y, A remark on the Bergman completeness, Complex Variables, 42, 11-15 (2000) · Zbl 1026.32060 · doi:10.1080/17476930008815267 |
| [13] | Chen, B-Y, Bergman completeness of hyperconvex manifolds, Nagoya Math. J., 175, 165-170 (2004) · Zbl 1061.32010 |
| [14] | Chen, B-Y, An essay on Bergman completeness, Ark. Mat., 51, 269-291 (2013) · Zbl 1276.32011 · doi:10.1007/s11512-012-0174-8 |
| [15] | Chen, B.-Y.: The Bergman metric and Isoperimetric inequalities, preprint |
| [16] | Chen, B-Y; Fu, S., Comparison of the Bergman and Szegö kernels, Adv. Math., 228, 2366-2384 (2011) · Zbl 1229.32008 · doi:10.1016/j.aim.2011.07.013 |
| [17] | Chen, B-Y; Kamimoto, J.; Ohsawa, T., Behavior of the Bergman kernel at infinity, Math. Z., 248, 695-708 (2004) · Zbl 1066.32003 · doi:10.1007/s00209-004-0676-6 |
| [18] | Chen, B-Y; Zhang, J-H, The Bergman metric on a Stein manifold with a bounded plurisubharmonic function, Trans. Am. Math. Soc., 354, 2997-3009 (2002) · Zbl 0997.32011 · doi:10.1090/S0002-9947-02-02989-6 |
| [19] | Claudon, B.: Non algebraicity of universal covers of Kähler surfaces. arXiv:1001.2379v1 |
| [20] | Conway, J.B.: Functions of One Complex Variable II (GTM, vol. 159), Spinger, New York (1995) · Zbl 0887.30003 |
| [21] | Demailly, J-P, Mesures de Monge-Ampere et mesures pluriharmoniques, Math. Z., 194, 519-564 (1987) · Zbl 0595.32006 · doi:10.1007/BF01161920 |
| [22] | Diederich, K.; Ohsawa, T., An estimate for the Bergman distance on pseudoconvex domains, Ann. Math., 141, 181-190 (1995) · Zbl 0828.32002 · doi:10.2307/2118631 |
| [23] | Donnelly, H.; Fefferman, C., \(L^2\) -cohomology and index theorem for the Bergman metric, Ann. Math., 118, 593-618 (1983) · Zbl 0532.58027 · doi:10.2307/2006983 |
| [24] | Greene, R.E., Wu, H.: Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Mathematics, vol. 699. Springer, Berlin (1979) · Zbl 0414.53043 |
| [25] | Harvey, FR; Wells, RO, Holomorphic approximation and hyperfunction theory on \(C^1\) totally real submanifold of a complex manifold, Math. Ann., 197, 287-318 (1972) · Zbl 0246.32019 · doi:10.1007/BF01428202 |
| [26] | Herbort, G., The Bergman metric on hyperconvex domains, Math. Z., 232, 183-196 (1999) · Zbl 0933.32048 · doi:10.1007/PL00004754 |
| [27] | Herbort, G., The pluricomplex Green function on pseudoconvex domains with a smooth boundary, Int. J. Math., 11, 509-522 (2000) · Zbl 1110.32307 |
| [28] | Hörmander, L.: An introduction to Complex Analysis in Several Variables, North Holland, Amsterdam (1990) · Zbl 0685.32001 |
| [29] | Jarnicki, M.; Pflug, P., Bergman completeness of complete circular domains, Ann. Pol. Math., 50, 219-222 (1989) · Zbl 0701.32002 |
| [30] | Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis, De Gruyter Expositions in Math., vol. 9. Walter de Gruyter & Co., Berlin(1993) · Zbl 0789.32001 |
| [31] | Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis-Revisited, Dissertationes Math., vol. 430, 192 pp (2005) · Zbl 1085.32005 |
| [32] | Kanai, M., Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Jpn., 37, 391-413 (1985) · Zbl 0554.53030 · doi:10.2969/jmsj/03730391 |
| [33] | Kerzman, N.; Rosay, JP, Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut, Math. Ann., 257, 171-184 (1981) · Zbl 0451.32012 · doi:10.1007/BF01458282 |
| [34] | Kobayashi, S., Geometry of bounded domains, Trans. Am. Math. Soc., 92, 267-290 (1959) · Zbl 0136.07102 · doi:10.1090/S0002-9947-1959-0112162-5 |
| [35] | Kobayashi, S., On complete Bergman metrics, Proc. Am. Math. Soc., 13, 511-513 (1962) · Zbl 0114.04001 · doi:10.1090/S0002-9939-1962-0141795-0 |
| [36] | Kobayashi, S.: Hyperbolic Complex spaces, A Series of Comprehensive Studies in Mathematics, vol. 318, Springer, Berlin (1998) · Zbl 0917.32019 |
| [37] | Krushkal, SL, Strengthening pseudoconvexity of finite-dimentional Teichmüller spaces, Math. Ann., 290, 681-687 (1991) · Zbl 0702.30049 · doi:10.1007/BF01459267 |
| [38] | Nikolov, N.; Pflug, P.; Zwonek, W., Estimates for invariant metrics on \({\mathbb{C}}-\) convex domains, Trans. Am. Math. Soc., 363, 6245-6256 (2011) · Zbl 1232.32005 · doi:10.1090/S0002-9947-2011-05273-6 |
| [39] | Ohsawa, T., A remark on the completeness of the Bergman metric, Proc. Jpn. Acad., 57, 238-240 (1981) · Zbl 0508.32008 · doi:10.3792/pjaa.57.238 |
| [40] | Ohsawa, T.: Boundary behavior of the Bergman kernel function on pseudoconvex domains, vol. 20, pp. 897-902. Publ. RIMS, Kyoto Univ. (1984) · Zbl 0569.32013 |
| [41] | Ohsawa, T., On the Bergman kernel of hyperconvex domains, Nagoya Math. J., 129, 43-52 (1993) · Zbl 0774.32016 |
| [42] | Ohsawa, T., Addendum to “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J., 137, 145-148 (1995) · Zbl 0817.32013 |
| [43] | Ohsawa, T.; Sibony, N., Bounded p.s.h. functions and pseudoconvexity in Kähler manifolds, Nagoya Math. J., 149, 1-8 (1998) · Zbl 0911.32027 |
| [44] | Pflug, P., Quadratintegrable holomorphe Functionen und die Serre-Vermutung, Math. Ann., 216, 285-288 (1975) · Zbl 0294.32009 · doi:10.1007/BF01430969 |
| [45] | Pflug, P.; Zwonek, W., Bergman completeness of unbounded Hartogs domains, Nagoya Math. J., 180, 121-133 (2005) · Zbl 1094.32004 |
| [46] | Pflug, P.; Zwonek, W., Logarithmic capacity and Bergman functions, Arch. Math., 80, 536-552 (2003) · Zbl 1037.30009 · doi:10.1007/s0013-003-0096-6 |
| [47] | Rosay, J-P; Rudin, W., Holomorphic maps from \({\mathbb{C}}^n\) to \({\mathbb{C}}^n\), Trans. Am. Math. Soc., 310, 47-86 (1988) · Zbl 0708.58003 |
| [48] | Siu, Y-T, Every Stein subvariety admits a Stein neighborhood, Invent. Math., 38, 89-100 (1976) · Zbl 0343.32014 · doi:10.1007/BF01390170 |
| [49] | Sugawa, T.: Uniformly perfect sets: analytic and geometric aspects, Sugaku Expositions (2003) |
| [50] | \(V \check{\rm a}\) j \(\check{\rm a}\) itu, V.: On locally hyperconvex morphisms, C. R. Acad. Sci. Paris Sér. I Math. 322, 823-828 (1996) · Zbl 0866.32008 |
| [51] | Wang, X., Bergman completeness is not a quasiconformal invariant, Proc. Am. Math. Soc., 141, 543-548 (2003) · Zbl 1264.32006 · doi:10.1090/S0002-9939-2012-11328-0 |
| [52] | Wolpert, S.A.: Weil-Petersson perspectives, Proc. Sympos. Pure Math., vol. 74, Am. Math. Soc., Procidence, RI, 2006 · Zbl 1259.32004 |
| [53] | Yau, S-T, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Math. J., 25, 659-670 (1976) · Zbl 0335.53041 · doi:10.1512/iumj.1976.25.25051 |
| [54] | Zwonek, W., An example concerning Bergman completeness, Nagoya Math. J., 164, 89-101 (2001) · Zbl 1038.32014 |
| [55] | Zwonek, W., Wiener’s type criterion for Bergman exhaustiveness, Bull. Pol. Acad. Sci. Math., 50, 297-311 (2002) · Zbl 1016.32004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
