We show that for any
C
0
\mathcal {C}^0
Jordan curve
Γ
\Gamma
in
S
∞
2
(
H
3
)
S^2_{\infty }(\mathbf {H}^3)
, there exists an embedded
H
H
-plane
P
H
\mathcal {P}_H
in
H
3
\mathbf {H}^3
with
∂
∞
P
H
=
Γ
\partial _{\infty } \mathcal {P}_H =\Gamma
for any
H
∈
(
−
1
,
1
)
H\in (-1,1)
. As a corollary, we prove that any quasi-Fuchsian hyperbolic
3
3
-manifold
M
≃
Σ
×
R
M\simeq \Sigma \times \mathbb {R}
contains an
H
H
-surface
Σ
H
\Sigma _H
in the homotopy class of the core surface
Σ
\Sigma
for any
H
∈
(
−
1
,
1
)
H\in (-1,1)
. We also prove that for any
C
1
C^1
Jordan curve in
S
∞
2
(
H
3
)
S^2_{\infty }(\mathbf {H}^3)
, there exists a unique minimizing
H
H
-plane
P
H
\mathcal {P}_H
with
∂
∞
P
H
=
Γ
\partial _{\infty } \mathcal {P}_H =\Gamma
for a generic
H
∈
(
−
1
,
1
)
H\in (-1,1)
.