Appendix A The Biholomorphism for the \(2\)-Sphere

Recall that the Killing form \(B\) on \(\mathfrak{su}(2)\) is negative definite. Therefore, fixing an isomorphism \((\mathfrak{su}(2),-B)\simeq \mathbb{E}^3\) with Euclidean \(3\)-space \(\mathbb{E}^3\), the adjoint representation of \(\mathrm{SU}(2)\) gives a group homomorphism \[\mathrm{Ad} : \mathrm{SU}(2)\to \mathrm{SO}(\mathfrak{su}(2),-B)\simeq \mathrm{SO}(3).\] Explicitly, mapping the \(-B\)-orthonormal basis of \(\mathfrak{su}(2)\) given by \[b_1=\frac{\sqrt{2}}{4}\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}\quad \text{and} \quad b_2=\frac{\sqrt{2}}{4}\begin{pmatrix}0 & \mathrm{i}\\ \mathrm{i}& 0\end{pmatrix}\quad \text{and} \quad b_3=\frac{\sqrt{2}}{4}\begin{pmatrix}-\mathrm{i}& 0 \\ 0 & \mathrm{i}\end{pmatrix}\] to the standard basis of \(\mathbb{R}^3\), the adjoint representation becomes \[\begin{gathered} \begin{pmatrix} z& -\overline w\\ w& \overline z\end{pmatrix}\mapsto \\ \frac{1}{2}\begin{pmatrix} \left(z^2+w^2+\overline{z}^2+\overline{w}^2\right) & -\mathrm{i}\left(z^2+w^2-\overline{z}^2-\overline{w}^2\right) & 2\mathrm{i}(z\overline{w}-\overline{z}w) \\ \mathrm{i}\left(z^2-w^2-\overline{z}^2+\overline{w}^2\right) & \left(z^2+\overline{z}^2-w^2-\overline{w}^2\right) & -2(z\overline{w}+\overline{z}w) \\ 2\mathrm{i}(zw-\overline{zw}) & 2(zw+\overline{zw}) & 2(|z|^2-|w|^2)\end{pmatrix}.\end{gathered}\] The unit tangent bundle of the Euclidean \(2\)-sphere \(S^2\subset \mathbb{E}^3\) may be identified with \(\mathrm{SO}(3)\) by thinking of the third column vector \(e\) of an element \((e_1\;e_2\;e)\in \mathrm{SO}(3)\) as the basepoint \(e\in S^2\) and by thinking of the remaining two column vectors \((e_1\;e_2)\) as a positively oriented orthonormal basis of \(T_{e}S^2\), where we identify the tangent plane to \(e\) with the orthogonal complement \(\{e\}^{\perp}\) of \(\mathrm{e}\) in \(\mathbb{E}^3\).

We can represent an orientation compatible linear complex structure \(J\) on \(T_{e}S^2\) by mapping \(J\) to the (complex) projectivisation of the vector \(v+iJv\), where \(v \in T_{e}S^2\simeq \{e\}^{\perp}\subset \mathbb{R}^3\) is any non-zero tangent vector. This defines a map \(\mathsf{J}_+(S^2) \to \mathbb{CP}^2\). Since \(v\) and \(Jv\) are linearly independent, the image of this map is disjoint from \(\mathbb{RP}^2\), where we think of \(\mathbb{RP}^2\) as sitting inside \(\mathbb{CP}^2\) via its standard real linear embedding. An elementary calculation shows that an orientation compatible linear complex structure \(J\) on \(T_{e}S^2\) with Beltrami coefficient \(\mu \in \mathbb{D}\) relative to the standard complex structure \(J_0\) has matrix representation \[J=\frac{1}{1-|\mu|^2}\begin{pmatrix}-2 \mathrm{Im}(\mu) & -|\mu|^2+2\mathrm{Re}(\mu)-1 \\ |\mu|^2+2\mathrm{Re}(\mu)+1 & 2\mathrm{Im}(\mu)\end{pmatrix}\] with respect to the basis \(\{e_1,J_0e_1\}\) of \(T_{e}S^2\). Hence, we obtain a map \[\begin{aligned} \mathrm{SO}(3) \times \mathbb{D} &\to \mathbb{CP}^2\setminus \mathbb{RP}^2 \\ \left[\left(e_1,e_2,e\right),\mu\right] &\mapsto \left[e_1+\frac{\mathrm{i}}{1-|\mu|^2}\left(-2\mathrm{Im}(\mu)e_1+(|\mu|^2+2\mathrm{Re}(\mu)+1)e_2\right)\right],\end{aligned}\] where we have used that the standard complex structure \(J_0\) of \(S^2\) satisfies \(e_2=J_0e_1\) for all \((e_1\;e_2\;e) \in \mathrm{SO}(3)\). Composing with the adjoint representation this becomes the map \(\mathrm{SU}(2) \times \mathbb{D} \to \mathbb{CP}^2\setminus \mathbb{RP}^2\) defined by \[\begin{gathered} \left[\left(z,w\right),\mu\right] \mapsto \\ \left[z^2+w^2-\mu(\overline{z}^2+\overline{w}^2):\mathrm{i}\left(z^2-w^2+\mu(\overline{z}^2-\overline{w}^2)\right):2\mathrm{i}(zw+\mu\overline{zw})\right]. \end{gathered}\] A linear coordinate transformation \[\mathbb{C}^3\to \mathbb{C}^3, \quad (z_1,z_2,z_3) \mapsto (\mathrm{i}z_1+ z_2,z_3,\mathrm{i}z_1-z_2)\] thus gives the map \[\mathrm{SU}(2)\times \mathbb{D} \to \mathbb{CP}^2, \quad [(z,w),\mu] \mapsto [z^2-\mu \overline{w}^2 : zw+\mu\overline{zw}:w^2-\mu\overline{z}^2]\] used in 4.4.