拟凸Hartogs域到复空间形式的全纯等距嵌入映射的存在性

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1511-1524.

拟凸Hartogs域到复空间形式的全纯等距嵌入映射的存在性

郝毅红^1,*(),王安²()

¹ 西北大学数学学院西安 710127
² 首都师范大学数学科学学院北京 100048

收稿日期:2019-05-24 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 郝毅红 E-mail:haoyihong@nwu.edu.cn;wangan@mail.cnu.edu.cn
作者简介:王安, E-mail:wangan@mail.cnu.edu.cn
基金资助:
国家自然科学基金(11601422);国家自然科学基金(11671270);国家自然科学基金(11871044);北京市自然科学基金(1182008);河北省自然科学基金(A2019106037);陕西省科技厅自然科学基础研究计划(2019JQ-398);陕西省教育厅专项科学研究计划(19JK0841);西北大学自然科学基金(15NW23)

Kähler Immersions of Pseudoconvex Hartogs Domains into Complex Space Forms

Yihong Hao^1,*(),An Wang²()

¹ Department of Mathematics, Northwest University, Xi'an 710127
² School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received:2019-05-24 Online:2020-12-26 Published:2020-12-29
Contact: Yihong Hao E-mail:haoyihong@nwu.edu.cn;wangan@mail.cnu.edu.cn
Supported by:
the NSFC(11601422);the NSFC(11671270);the NSFC(11871044);the Beijing Municipal Natural Science Foundation(1182008);the NSF of Hebei Province(A2019106037);the NSF of Shannxi Province(2019JQ-398);the Scientific Research Program of Shaanxi Provincial Education Department(19JK0841);the Science Research Foundation of Northwest University(15NW23)

摘要/Abstract

摘要：

在${\Bbb C}^{n}$中的有界拟凸Hartogs域$\Omega$上, 存在一个自然的Kähler度量$g^{\Omega}$.该文将研究有界拟凸Hartogs域$(\Omega, h g^{\Omega})$到复空间形式的全纯等距嵌入映射的存在性问题.

关键词: 全纯等距嵌入映射, 复空间形式, 拟凸Hartogs域

Abstract:

Let $\Omega$ be a bounded pseudoconvex Hartogs domain in ${\Bbb C}^{n}$. There exists a natural Kähler metric $g^{\Omega}$ in terms of its defining function. In this paper, we study the existence of Kähler immersions of $(\Omega, h g^{\Omega})$ into finite and infinite dimensional complex space forms for $h>0$.

Key words: Kähler immersion, Complex space form, Pseudoconvex Hartogs domain

中图分类号:

O174.5

郝毅红,王安. 拟凸Hartogs域到复空间形式的全纯等距嵌入映射的存在性[J]. 数学物理学报, 2020, 40(6): 1511-1524.

Yihong Hao,An Wang. Kähler Immersions of Pseudoconvex Hartogs Domains into Complex Space Forms[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1511-1524.

参考文献 28

1	Bi E , Feng Z , Tu Z . Balanced metrics on the Fock-Bargmann-Hartogs domains. Ann Glob Anal Geom, 2016, 49: 349- 359
2	Calabi E . Isometric imbedding of complex manifolds. Ann of Math, 1953, 58: 1- 23
3	Cheng X L , Niu Y Y . Submanifolds of Cartan-Hartogs domains and complex Euclidean spaces. J Math Anal Appl, 2017, 452 (2): 1262- 1268
4	Cheng X L , Di Scala A , Yuan Y . Käahler submanifolds and the Umehara algebra. Int J Math, 2017, 28 (4): 1750027
5	Cheng X L, Hao Y H. Non-relativity of Käahler manifold and complex space forms. 2005, arXiv: 2005.03208
6	Chern S S . On Einstein hypersurfaces in a Käahler manifold of constant bisectional curvature. J Differ Geom, 1967, 1: 21- 31
7	Di Scala A J , Loi A . Käahler maps of Hermitian symmetric spaces into complex space forms. Geom Dedicata, 2007, 125: 103- 113
8	Di Scala A J , Loi A . Käahler manifolds and their relatives. Ann Sc Norm Super Pisa Cl Sci, 2010, 9 (5): 495- 501
9	Di Scala A J , Ishi H , Loi A . Käahler immersions of homogeneous Käahler manifolds into complex space forms. Asian J of Math, 2012, 16 (3): 479- 488
10	Hao Y H , Wang A . The Bergman kernels of generalized Bergman-Hartogs domains. J Math Anal Appl, 2015, 429 (1): 326- 336
11	Huang X, Yuan Y. Submanifolds of Hermitian symmetric spaces//Baklouti A, Kacimi A, Kallel S, Mir N. Analysis and Geometry. Heidelberg: Springer, 2015: 197-206
12	Hulin D . Sous-variétés complexes d'Einstein de l'espace projectif (France). Bull Soc Math, 1996, 124: 277- 298
13	Hulin D . Käahler-Einstein metrics and projective embeddings. J Geom Anal, 2000, 10 (3): 525- 528
14	Ishi H , Park J D , Yamamori A . Bergman kernel function for Hartogs domains over bounded homogeneous domains. J Geom Anal, 2017, 27: 1703- 1736
15	Kim H , Ninh V T , Yamamori A . The automorphism group of a certain unbounded non-hyperbolic domain. J Math Anal Appl, 2014, 409 (2): 637- 642
16	Kim H , Yamamori A . An application of a Diederich-Ohsawa theorem in characterizing some Hartogs domains. Bull Sci Math, 2015, 139 (7): 737- 749
17	Loi A , Zedda M . Käahler-Einstein submanifolds of the infnite dimensional projective space. Math Ann, 2011, 350 (1): 145- 154
18	Loi A , Mossa R . Some remarks on homogeneous Käahler manifolds. Geom Ded, 2015, 179: 377- 383
19	Loi A . Holomorphic maps of Hartogs domains in complex space forms. Riv Mat Univ Parma, 2002, 7: 103- 113
20	Loi A , Zedda M . Käahler Immersions of Käahler Manifolds into Complex Space Forms. Cham: Springer, 2018
21	Mossa R . A bounded homogeneous domain and a projective manifold are not relatives. Riv Mat Univ Parma, 2013, 4 (1): 55- 59
22	Su G , Tang Y Y , Tu Z H . Käahler submanifolds of the symmetrized polydisc. C R Acad Sci Paris Ser I, 2018, 356: 387- 394
23	Tsukada K . Einstein-Käahler submanifolds with codimension two in a complex space form. Math Ann, 1986, 274: 503- 516
24	Umehara M . Einstein-Käahler submanifolds of complex linear or hyperbolic space. Tohoku Math J, 1987, 39 (3): 385- 389
25	Umehara M . Käahler submanifolds of complex space forms. Tokyo J Math, 1987, 10 (1): 203- 214
26	Zhao J , Wang A , Hao Y H . On the holomorphic automorphism group of the Bergman-Hartogs domain. Int J Math, 2015, 26 (8): 1550056
27	Zedda M. Käahler Immersions of Käahler-Einstein Manifolds into Infinite Dimensional Complex Space Form[D]. Cagliari: Università degli Studi di Cagliari, 2009
28	Zedda M. Strongly not relative Käahler manifolds. Complex Manifolds, 2016, arXiv: 1608.03163