On canonical metrics on Cartan–Hartogs domains
The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain Ω B d 0 ( μ ) endowed with the canonical metric g ( μ ) , we obtain an explicit formula for the Bergman ker...
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| Veröffentlicht in: | Mathematische Zeitschrift Jg. 278; H. 1-2; S. 301 - 320 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.10.2014
|
| Schlagworte: | |
| ISSN: | 0025-5874, 1432-1823 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain
Ω
B
d
0
(
μ
)
endowed with the canonical metric
g
(
μ
)
,
we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space
H
α
of square integrable holomorphic functions on
Ω
B
d
0
(
μ
)
,
g
(
μ
)
with the weight
exp
{
-
α
φ
}
(where
φ
is a globally defined Kähler potential for
g
(
μ
)
) for
α
>
0
, and, furthermore, we give an explicit expression of the Rawnsley’s
ε
-function expansion for
Ω
B
d
0
(
μ
)
,
g
(
μ
)
.
Secondly, using the explicit expression of the Rawnsley’s
ε
-function expansion, we show that the coefficient
a
2
of the Rawnsley’s
ε
-function expansion for the Cartan–Hartogs domain
Ω
B
d
0
(
μ
)
,
g
(
μ
)
is constant on
Ω
B
d
0
(
μ
)
if and only if
Ω
B
d
0
(
μ
)
,
g
(
μ
)
is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda. |
|---|---|
| ISSN: | 0025-5874 1432-1823 |
| DOI: | 10.1007/s00209-014-1316-4 |