Abstract
We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$
A
=
{
(
z
,
w
)
∈
C
2
:
v
>
F
(
z
,
u
)
}
, where $$w = u + iv$$
w
=
u
+
i
v
and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$
C
z
×
R
u
, the following conditions are equivalent:
The domain $$\mathfrak {A}$$
A
is Kobayashi hyperbolic.
The domain $$\mathfrak {A}$$
A
is Brody hyperbolic.
The domain $$\mathfrak {A}$$
A
possesses a Bergman metric.
The domain $$\mathfrak {A}$$
A
possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$
c
(
A
)
of $$\mathfrak {A}$$
A
is empty.
The graph $$\Gamma (F)$$
Γ
(
F
)
of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$
Γ
(
H
)
of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$
H
:
C
z
→
C
w
.
A REMARK ON KAZHDAN'S THEOREM ON SEQUENCES OF BERGMAN METRICS
