Minimal immersions of the 2-sphere

Question

Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in his 1972 paper. However, his construction seems to be very unclear because of heavy use of coördinates. Are there any more coördinate-free expositions of this topic?

Answer 1

I don't know exactly where it appears in the literature, but a coordinate-free interpretation is certainly known, based on the idea of Penrose's twistor construction. The main points can be summarized as follows:

First, everything is equivariant under the group $\mathrm{SO}(2m{+}1)$: One has $S^{2m} = \mathrm{SO}(2m{+}1)/\mathrm{SO}(2m)$ as the space of unit vectors $\mathbb{R}^{2m+1}$, and $\mathrm{Gr}^+_2(\mathbb{R}^{2m+1})$, the Grassmannian of oriented $2$-planes in $\mathbb{R}^{2m+1}$ is $\mathrm{SO}(2m{+}1)/\bigl(\mathrm{SO}(2){\times}\mathrm{SO}(2m-1)\bigr)$. This latter space can be realized as the complex quadric $Q_{2m-1}\subset\mathbb{CP}_{2m}$ by identifying a unit bi-vector $e_1\wedge e_2$ (where $(e_1,e_2)$ is an oriented orthonormal basis of the $2$-plane $P\in\mathrm{Gr}^+_2(\mathbb{R}^{2m+1})$) with the 'isotropic' complex line $[e_1{-}ie_2]\in\mathbb{CP}_{2m}$. (It is called 'isotropic' because it is spanned by the complex vector $v = e_1{-}ie_2$ that satisfies $v\cdot v = 0$, i.e., $v$ is 'null', aka 'isotropic'.) It is easy to see that the correspondence $P\leftrightarrow [e_1{-}ie_2]$ is one-to-one and onto.

It's also worthwhile to note that a maximal torus $T^m$ in $\mathrm{SO}(2m)$ is also a maximal torus in $\mathrm{SO}(2m{+}1)$ and can be understood as the stabilizer of a splitting of $\mathbb{R}^{2m}$ as an orthogonal direct sum of $m$ oriented $2$-planes. In particular, the space $\mathrm{SO}(2m{+}1)/T^m$ can be thought of as the space of orthogonal splittings $\mathbb{R}^{2m+1} = L \oplus P_1\oplus\cdots\oplus P_m$, where $L$ is an oriented line (i.e., an element of $S^{2m}$) and each $P_i$ is an oriented $2$-plane (i.e., an element of $\mathrm{Gr}^+_2(\mathbb{R}^{2m+1})$). Note that, since $\mathrm{SO}(2m{+}1)/T^m$ is a compact simple group modulo a maximal torus, it has a natural complex structure. In fact, it has more than one, but there is a particularly nice one to choose for applications to this problem, one such that the mapping $p: \mathrm{SO}(2m{+}1)/T^m\to Q_{2m-1}\subset\mathbb{CP}_{2m}$ given by $$ p\bigl(L \oplus P_1\oplus\cdots\oplus P_m) = \overline{P_m} $$ is holomorphic.

Next, back in the 60s, Calabi had shown that, if $x:S^2\to S^{2m}$ is a (branched) minimal immersion of the $2$-sphere that is non-degenerate (i.e., its image does not lie in a great $(2m{-}1)$-sphere), then $x$ has a canonical lifting $\hat x:S^2\to\mathrm{SO}(2m{+}1)/T^m$ as a holomorphic rational curve of the form $$ \hat x = \mathbb{R}\,x \oplus \pi_1(x)\oplus \cdots \oplus \pi_m(x) $$ Basically, the idea is that one chooses a splitting of the $x$-pullback of the tangent bundle of $S^{2m}$ into oriented $2$-planes corresponding to the osculating spaces of the immersion. For example, $\pi_1(x)$ represents the (oriented) tangent space to the image, $\pi_2(x)$ represents the first normal space to the image, etc.

What Barbosa observed is that the map $$ p(\hat x):S^2\to \mathrm{Gr}^+_2(\mathbb{R}^{2m+1}) = Q_{2m-1}\subset \mathbb{CP}_{2m} $$ is holomorphic, and he showed that there is a way to recover $x$ from $p(\hat x)$.

Moreover, he showed that, if one starts with a holomorphic rational curve $\xi:S^2\to Q_{2m-1}\subset \mathbb{CP}_{2m}$ that is nondegenerate in the sense that its image does not lie in a hyperplane in $\mathbb{CP}_{2m}$, then it is of the form $\xi = p(\hat x)$ for some unique branched minimal immersion $x:S^2\to S^{2m}$.