### Abstract

The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in C^{2} is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ω_{m} = {(z_{1}, Z_{2}); [Z_{1}]^{2} + [Z_{2}]^{2m} < 1} or a tube domain T_{m} = {(z_{1}, Z_{2}); Imz_{1} + (Imz_{2})^{2m} < 1}. The Bergman metric for the tube domain T_{m} is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domain T_{m} at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kahler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in C^{2}.

Original language | English (US) |
---|---|

Journal | Journal of Geometric Analysis |

Volume | 6 |

Issue number | 3 |

State | Published - Dec 1 1996 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Keywords

- Bergman metric
- Finite type
- Pseudoconvex
- Reinhardt domain
- Tube domain

### Cite this

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^{2}',

*Journal of Geometric Analysis*, vol. 6, no. 3.

**Geometry of reinhardt domains of finite type in C ^{2}.** / Fu, Siqi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Geometry of reinhardt domains of finite type in C2

AU - Fu, Siqi

PY - 1996/12/1

Y1 - 1996/12/1

N2 - The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in C2 is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ωm = {(z1, Z2); [Z1]2 + [Z2]2m < 1} or a tube domain Tm = {(z1, Z2); Imz1 + (Imz2)2m < 1}. The Bergman metric for the tube domain Tm is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domain Tm at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kahler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in C2.

AB - The asymptotic behavior of the holomorphic sectional curvature of the Bergman metric on a pseudoconvex Reinhardt domain of finite type in C2 is obtained by rescaling locally the domain to a model domain that is either a Thullen domain Ωm = {(z1, Z2); [Z1]2 + [Z2]2m < 1} or a tube domain Tm = {(z1, Z2); Imz1 + (Imz2)2m < 1}. The Bergman metric for the tube domain Tm is explicitly calculated by using Fourier-Laplace transformation. It turns out that the holomorphic sectional curvature of the Bergman metric on the tube domain Tm at (0, 0) is bounded above by a negative constant. These results are used to construct a complete Kahler metric with holomorphic sectional curvature bounded above by a negative constant for a pseudoconvex Reinhardt domain of finite type in C2.

KW - Bergman metric

KW - Finite type

KW - Pseudoconvex

KW - Reinhardt domain

KW - Tube domain

UR - http://www.scopus.com/inward/record.url?scp=8744241630&partnerID=8YFLogxK

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M3 - Article

VL - 6

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 3

ER -

^{2}. Journal of Geometric Analysis. 1996 Dec 1;6(3).