We consider the problem of minimizing the energy of the maps
u
(
r
,
θ
)
u(r,\theta )
from the annulus
Ω
ρ
=
B
1
∖
B
¯
ρ
\Omega _\rho =B_1\backslash \bar B_\rho
to
S
2
S^2
such that
u
(
r
,
θ
)
u(r,\theta )
is equal to
(
cos
θ
,
sin
θ
,
0
)
(\cos \theta ,\sin \theta ,0)
for
r
=
ρ
r=\rho
, and to
(
cos
(
θ
+
θ
0
)
(\cos (\theta +\theta _0)
,
sin
(
θ
+
θ
0
)
,
0
)
\sin (\theta +\theta _0),0)
for
r
=
1
r=1
, where
θ
0
∈
[
0
,
π
]
\theta _0\in [0,\pi ]
is a fixed angle. We prove that the minimum is attained at a unique harmonic map
u
ρ
u_\rho
which is a planar map if
log
2
ρ
+
3
θ
0
2
≤
π
2
\log ^2\rho +3\theta _0^2\le \pi ^2
, while it is not planar in the case
log
2
ρ
+
θ
0
2
>
π
2
\log ^2\rho +\theta _0^2>\pi ^2
. Moreover, we show that
u
ρ
u_\rho
tends to
v
¯
\bar v
as
ρ
→
0
\rho \to 0
, where
v
¯
\bar v
minimizes the energy of the maps
v
(
r
,
θ
)
v(r,\theta )
from
B
1
B_1
to
S
2
S^2
, with the boundary condition
v
(
1
,
θ
)
=
(
cos
(
θ
+
θ
0
)
v(1,\theta )=(\cos (\theta +\theta _0)
,
sin
(
θ
+
θ
0
)
,
0
)
\sin (\theta +\theta _0),0)
.