Appendix A Deriving the transport equation

Here we sketch how to derive the transport equation for the function \(u\) starting from the PDE \[\mathrm{D}^{\prime\prime}\mu-\mu\,\mathrm{D}^{\prime}\mu=\Phi\mu^3+\overline{\Phi}\] for the Beltrami differential \(\mu\). Let \((g,A,\theta)\) be the triple encoding \(\mathfrak{p}\) so that the connection form of \(\mathrm{D}\) on \(SM\) is (see (2.8)) \(\kappa=i\psi-2\theta_1\omega\), where we write \(\theta_1=\frac{1}{2}(\theta-iV\theta)\). Moreover, on \(SM\) the section \(\Phi\) of \(K^2\otimes \overline{K^*}\) is represented by \(a_3=\frac{1}{3}Va+ia\), where \(a(v)=\operatorname{Re}A(Jv,Jv,Jv)\), \(v \in SM\). Writing \(\mu_{-2}\) for the complex-valued function on \(SM\) representing the Beltrami differential \(\mu\) and \(\mu_2=\overline{\mu_{-2}}\), the PDE for \(\mu\) is equivalent to \[d\mu_{-2}=\mu_{-2}^{\prime}\omega+\left(\mu_{-2}\mu_{-2}^{\prime}+a_3\mu_{-2}^3+\overline{a_3}\right)\overline{\omega}+\overline{\kappa}\mu_{-2}-\kappa\mu_{-2},\] where \(\mu_{-2}^{\prime}\) is a complex-valued function on \(SM\). Since \(\mu_{-2}\) represents a section of \(\overline{K}\otimes K^*\simeq K^{-2}\), writing \(\eta_{\pm}=\frac{1}{2}\left(X\mp iH\right)\) we also have \[d\mu_{-2}=\eta_+(\mu_{-2})\omega+\eta_{-}(\mu_{-2})\overline{\omega}-2 i\mu_{-2} \psi.\] Thus the PDE is equivalent to the system \[\tag{A.1} \eta_{-}\mu_{-2}-\mu_{-2}\eta_+\mu_{-2}=a_3\mu_{-2}^3-2\mu_{-2}^2\theta_1-2\mu_{-2}\overline{\theta_{1}}+\overline{a_3}\] and \(V\mu_{-2}=-2i\mu_{-2}\). The Beltrami differential does only define a conformal equivalence class \([\hat{g}]\). We may fix a metric \(\hat{g}\in [\hat{g}]\) by requiring \[\frac{1}{2}\left(p+q\right)=\frac{1+|\mu_2|^2}{\left(1-|\mu_2|^2\right)^4},\] where again we specify the metric \(\hat{g}\) in terms of the functions \(p,q,r\). Explicitly, we have \[\frac{1}{2}(p-q)=\frac{\mu_{-2}+\mu_2}{\left(1-|\mu_2|^2\right)^4} \quad \text{and} \quad r=\frac{i(\mu_2-\mu_{-2})}{\left(1-|\mu_2|^2\right)^4}.\] In particular, this yields \[h:=\frac{p}{(pq-r^2)^{2/3}}=(\mu_{-2}+1)(\mu_2+1).\] Writing \(F=X+(a-V\theta)V\) and using (A.1), a lengthy but straightforward calculation shows that \[Fh=\frac{2}{3}hVa+2h\operatorname{Re}\left(a_3\mu_{-2}^2-\mu_2a_{-3}-2\mu_2\theta_{-1}+\eta_+\mu_{-2}\right).\] Hence if we define \(u=\frac{3}{2}\log h\), then we obtain \[Fu-Va=3\operatorname{Re}\left(a_3\mu_{-2}^2-\mu_2a_{-3}-2\mu_2\theta_{-1}+\eta_+\mu_{-2}\right)\] Note that the right hand side of the last equation lies in \(\mathcal{H}_{-1}\oplus \mathcal{H}_1\), hence there exists a \(1\)-form \(\beta\) on \(M\) so that \[Fu=Va+\beta\] which is the transport equation (5.5).