Genus zero Willmore surfaces immersed in the three-sphere
S
3
\mathbb {S}^3
correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are
4
π
k
4\pi k
, where
k
∈
N
∗
k\in \mathbb {N}^*
, with
k
≠
2
,
3
,
5
,
7
k\ne 2,3,5,7
. When the ambient space is the four-sphere
S
4
\mathbb {S}^4
, the regular homotopy class of immersions of the two-sphere
S
2
\mathbb {S}^2
is determined by the self-intersection number
q
∈
Z
q\in \mathbb {Z}
; here we shall prove that the possible critical values are
4
π
(
|
q
|
+
k
+
1
)
4\pi (|q|+k+1)
, where
k
∈
N
k\in \mathbb {N}
. Moreover, if
k
=
0
k=0
, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration
P
3
→
S
4
\mathbb {P}^3\rightarrow \mathbb {S}^4
, from a rational curve in
P
3
\mathbb {P}^3
and, if
k
≠
0
k\ne 0
, via stereographic projection, from a minimal surface in
R
4
\mathbb {R}^4
with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some
P
2
⊂
P
3
\mathbb {P}^2\subset \mathbb {P}^3
or (equivalently) when the minimal surface of
R
4
\mathbb {R}^4
is complex with respect to a suitable complex structure of
R
4
\mathbb {R}^4
.