4 The variational equations

Proof. First recall that in Lemma 3.3 we have defined \[\zeta_1=\theta^1_0+\mathrm{i}\theta^2_0,\quad \zeta_2=\left(\theta^1_1-\theta^2_2\right)+\mathrm{i}\left(\theta^1_2+\theta^2_1\right),\quad \zeta_3=\theta^0_1+\mathrm{i}\theta^0_2,\] where \(\theta=(\theta^i_j)\) is the Cartan connection of \((\Sigma,\mathfrak{p})\).

Let now \([g] : \Sigma \to Z\) be a conformal structure on \((\Sigma,\mathfrak{p})\) and let \(p : P_{[g]}=[g]^*P\to\Sigma\) denote the pullback of the bundle \(\mu : P\to Z\), that is, \[P_{[g]}=\left\{(p,u) \in \Sigma\times P\,|\, [g](p)=\mu(u)\right\}.\] Since \(P_{[g]}\) is \(6\)-dimensional, two of the components of \(\theta\) become linearly dependent when pulled back to \(P_{[g]}\). Clearly, these components must be among the \(1\)-forms that are semibasic for \(\mu\). Recall that these forms are spanned by \(\zeta_1,\zeta_2\) and their complex conjugates. However, since \([g]\) is a section of \(Z \to \Sigma\) and since the \(1\)-forms that are semibasic for the projection \(\pi : P\to \Sigma\) are spanned by \(\zeta_1,\overline{\zeta_1}\), it follows that \(\zeta_1\wedge\overline{\zeta_1}\) is non-vanishing on \(P_{[g]}\). Therefore, on \(P_{[g]}\) we have the relation \[\tag{4.2} \zeta_2=2\overline{a}\overline{\zeta_1}+c\zeta_1\] for unique complex-valued functions \(a,c\). From the equivariance properties of \(\zeta_1,\zeta_2\) under the \(\mathbb{R}_2\rtimes \mathrm{CO}(2)\)-right action (3.6), we obtain that for all \(u \in P_{[g]}\) and \(z \rtimes r\mathrm{e}^{\mathrm{i}\phi}\in \mathbb{R}_2\rtimes \mathrm{CO}(2)\) we have \[c(u\cdot z\rtimes r\mathrm{e}^{\mathrm{i}\phi})=r^3\mathrm{e}^{\mathrm{i}\phi}c(u)+ r^2 z\] and \[\tag{4.3} a(u\cdot z\rtimes r\mathrm{e}^{\mathrm{i}\phi})=r^3\mathrm{e}^{-3\mathrm{i}\phi}a(u).\] It follows that the equation \(c=0\) defines a locus that corresponds to a section \(\widetilde{[g]} : \Sigma \to Y\) covering \([g]\). On the pullback bundle \(P^{\prime}_{[g]}=\widetilde{[g]}^*P\), where \[P^{\prime}_{[g]}=\left\{(p,u) \in \Sigma\times P\,|\, \widetilde{[g]}(p)=\tau(u)\right\},\] we obtain \[\tag{4.4} \zeta_2=2\overline{a}\overline{\zeta_1}.\] Since \(P^{\prime}_{[g]}\) is \(4\)-dimensional, two of the remaining components of \(\theta\) become linearly dependent when pulled back to \(P^{\prime}_{[g]}\). Since the \(1\)-forms that are semibasic for the projection \(\tau : P\to Y\) are spanned by \(\zeta_1,\zeta_2,\zeta_3\) and their complex conjugates, it follows as before that \[\tag{4.5} \zeta_3=k\zeta_1+2\overline{q}\,\overline{\zeta_1}\] for unique complex-valued functions \(k,q\).

Now recall that Cartan’s bundle \(\pi : P\to \Sigma\) is isomorphic to \(F^+\times \mathbb{R}_2\to \Sigma\) equipped with the \(\mathrm{G}\)-right action (3.1). Therefore, \(P_{[g]} \to \Sigma\) is isomorphic to \(F^+_{[g]} \times \mathbb{R}_2 \to \Sigma\) and consequently, the bundle \(P^{\prime}_{[g]} \to \Sigma\) is isomorphic to \(F^+_{[g]} \to \Sigma\).

Proof. We will only verify the structure equation for \(a\) as the other structure equations are derived in an entirely analogous fashion. The structure equations (3.11) and (4.4) gives \[\begin{aligned} \mathrm{d}\overline{\zeta_2}&=\mathrm{d}(2a\zeta_1)=2\mathrm{d}a\wedge\zeta_1+a\,\mathrm{d}\zeta_1=-\zeta_1\wedge \mathrm{d}a +2a\left(\zeta_1\wedge \varphi+\frac{1}{2}\overline{\zeta_1}\wedge\zeta_2\right)\\ &=-\overline{\zeta_1}\wedge\overline{\zeta_3}+\overline{\zeta_2}\wedge \overline{\varphi}-\overline{\zeta_2}\wedge \varphi\\ &=2q\zeta_1\wedge\overline{\zeta_1}+2a\zeta_1\wedge\overline{\varphi}-2a\zeta_1\wedge \varphi, \end{aligned}\] where we have used (4.5). Equivalently, we obtain \[0=\left(\mathrm{d}a+q\overline{\zeta_1}-2a \varphi+a\overline{\varphi}\right)\wedge\zeta_1,\] which implies (4.6). Finally, the structure equation (4.7) for \(\varphi\) is an immediate consequence of (3.11), (4.4) and (4.5).

Proof. By construction of \(P^{\prime}_{[g]}\simeq F^+_{[g]}\), a complex-valued \(1\)-form on \(\Sigma\) is a \((1,\! 0)\)-form for the complex structure \(J\) induced by \([g]\) and the orientation if and only if its \(p\)-pullback to \(P^{\prime}_{[g]}\) is a complex multiple of \(\zeta_1\). Since \[\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*\zeta_1=\frac{1}{r^3}\mathrm{e}^{\mathrm{i}\phi}\zeta_1\] it follows that the sections of \(K_{\Sigma}^2\) are in one-to-one correspondence with the complex-valued functions \(f\) on \(P^{\prime}_{[g]}\) satisfying \[\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*f=r^3\mathrm{e}^{-\mathrm{i}\phi}r^3\mathrm{e}^{-\mathrm{i}\phi}f=r^6\mathrm{e}^{-2\mathrm{i}\phi} f.\] Likewise, it follows that the sections of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\) are in one-to-one correspondence with the complex-valued functions \(f\) on \(P^{\prime}_{[g]}\) satisfying \[\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*f=r^3\mathrm{e}^{-\mathrm{i}\phi}r^3\mathrm{e}^{-\mathrm{i}\phi}\overline{r^{-3}\mathrm{e}^{\mathrm{i}\phi}} f=r^3\mathrm{e}^{-3\mathrm{i}\phi}f\] and that the sections of \(K_{\Sigma}\otimes \overline{K_{\Sigma}}\) are in one-to-one correspondence with the complex valued functions \(f\) on \(P^{\prime}_{[g]}\) satisfying \[\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*f=r^3\mathrm{e}^{-\mathrm{i}\phi}r^3\mathrm{e}^{\mathrm{i}\phi}f=r^6f.\] From (4.5) and (3.9) we obtain that for all \(u \in P^{\prime}_{[g]}\) and \(r\mathrm{e}^{\mathrm{i}\phi}\in \mathrm{CO}(2)\) \[\begin{aligned} k(u\cdot r\mathrm{e}^{\mathrm{i}\phi})&=r^6k(u),\\ q(u\cdot r\mathrm{e}^{\mathrm{i}\phi})&=r^6\mathrm{e}^{-2\mathrm{i}\phi}q(u). \end{aligned}\] These equations imply that there exists a unique quadratic differential \(Q\) on \(\Sigma\) that is represented by \(q\) and a unique \((1,\! 1)\)-form \(\kappa\) on \(\Sigma\) that is represented by \(k\). Furthermore, (4.3) implies that there exists a unique section \(\alpha\) of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\) that is represented by \(a\).

It follows from the properties (ii) and (iii) of the Cartan connection that \(\varphi\) is a connection \(1\)-form on the \(\mathrm{CO}(2)\)-bundle \(P^{\prime}_{[g]} \to \Sigma\). Its pushforward under the bundle isomorphism \(P^{\prime}_{[g]} \to F^+_{[g]}\) is then a \(\mathrm{CO}(2)\)-connection on \(F^+_{[g]}\) which – by abuse of notation – we denote by \(\varphi\) as well. The structure equation (4.7) implies that \(\varphi\) is torsion-free.

Finally, the identity \(Q=-\nabla^{\prime\prime}_\varphi\alpha\) is an immediate consequence of the structure equation (4.6) and Lemma 4.1.

Proof of Corollary 4.6. By construction, the metric \(h_{\mathfrak{p}}\) has the property that its pullback to \(P\) is \[\tau^*h_{\mathfrak{p}}=\frac{1}{2}\left(\zeta_1\circ \overline{\zeta_3}+\zeta_3\circ\overline{\zeta_1}+\zeta_2\circ\overline{\zeta_2}\right).\] Therefore, from (4.4) and (4.5) it follows that \[\tag{4.9} p^*\left(\widetilde{[g]}^*h_{\mathfrak{p}}\right)=\frac{1}{2}\left(4|a|^2+(k+\overline{k})\right)\zeta_1\circ\overline{\zeta_1}+q\,\zeta_1\circ\zeta_1+\overline{q}\,\overline{\zeta_1}\circ\overline{\zeta_1}.\] Since a complex-valued \(1\)-form on \(\Sigma\) is a \((1,\! 0)\)-form for the complex structure defined by \([g]\) and the orientation if and only if its \(p\)-pullback to \(P^{\prime}_{[g]}\) is a complex multiple of \(\zeta_1\), equation (4.9) implies that \(\widetilde{[g]}^*h_{\mathfrak{p}}\) is weakly conformal to \([g]\) if and only if \(q\) vanishes identically. The first claim follows.

The second part of the claim is an immediate consequence of (4.4) and the characterisation of the complex structures on \(Z,Y\) in terms of \(\zeta_1,\zeta_2,\zeta_3\) and the characterisation of the holomorphic contact structure in terms of \(\zeta_2=0\).

Proof of Proposition 4.9. Without losing generality, we can assume that the projective structure \(\mathfrak{p}\) is defined by \({}^{[g]}\nabla+A_{[g]}\) for some \([g]\)-conformal connection \({}^{[g]}\nabla\) and some \(1\)-form \(A_{[g]}\) having all the properties of Theorem 2.4. Recall (3.3) that the choice of a representative connection \(\nabla \in \mathfrak{p}\) gives an identification \(P\simeq F^+\rtimes\mathbb{R}_2\) of Cartan’s bundle so that the Cartan connection form becomes \[\tag{4.17} \theta=\begin{pmatrix} -\frac{1}{3}\operatorname{tr}\eta-\xi \omega & \mathrm{d}\xi-\xi\eta-(S\omega)^t-\xi\omega\xi\\ \omega & \eta-\frac{1}{3}\mathrm{I}\operatorname{tr}\eta+\omega\xi \end{pmatrix}.\] We will construct Cartan’s connection for the representative connection \[\tag{4.18} {}^{(g,\beta)}\nabla+A_{[g]}={}^g\nabla+g\otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes \beta+A_{[g]}.\] Let \(A^i_{jk}\) denote the real-valued functions on \(F^+\) representing \(A_{[g]}\). In particular, we have \[\tag{4.19} A^i_{jk}=A^i_{kj}\quad \text{and} \quad A^l_{il}=0.\] On \(F^+\) the connection form of (4.18) is given by \[\tag{4.20} \eta^i_j=\psi^i_j+\left(b_kg^{ki}g_{jl}-\delta^i_jb_l-\delta^i_lb_j+A^i_{jl}\right)\omega^l,\] By definition, the pullback bundle \(P_{[g]}\) is the subbundle of \(F^+\times \mathbb{R}_2\) defined by the equations \(g_{11}=g_{22}\) and \(g_{12}=0\). Now on \(P_{[g]}\simeq F^+_{[g]}\times \mathbb{R}_2\) we have \(\psi^2_1=-\psi^1_2\) and \(\psi^1_1=\psi^2_2\). Using (4.17), (4.19) and (4.20) we compute \[\begin{aligned} \zeta_2&=(\theta^1_1-\theta^2_2)+\mathrm{i}\left(\theta^1_2+\theta^2_1\right)\\ &=\psi^1_1-\psi^2_2+\left(\xi_1+2A^1_{11}\right)\omega^1+\left(-\xi_2-2A^2_{22}\right)\omega^2\\ &\phantom{=}+\mathrm{i}\left(\psi^1_2+\psi^2_1+\left(\xi_2-2A^2_{22}\right)\omega^1+\left(\xi_1-2A^1_{11}\right)\omega^2\right)\\ &=2\overline{a}\overline{\zeta_1}+c\zeta_1, \end{aligned}\] where \[\tag{4.21} \begin{aligned} a&=A^1_{11}+\mathrm{i}A^2_{22},\\ c&=\xi_1+\mathrm{i}\xi_2 \end{aligned}\] and we have used that on \(F^+_{[g]}\) \[\delta_{il}A^l_{jk}=\delta_{jl}A^l_{ik},\] which follows from Theorem 2.4 (vi). Recall that \(P^{\prime}_{[g]}\) was defined by the equation \(c=0\). Hence on \(P^{\prime}_{[g]}\simeq F^+_{[g]}\) the function \(\xi\) vanishes identically. Using this we compute \[\varphi=-\frac{1}{2}\left(3\theta^0_0+\mathrm{i}\left(\theta^1_2-\theta^2_1\right)\right)=\psi^1_1-b_1\omega^1-b_2\omega^2+\mathrm{i}\left(\psi^2_1+b_2\omega^1-b_1\omega^2\right).\] This is precisely (4.16). It follows that the connection defined by \(\varphi\) is the same as the induced torsion-free connection on \(F^+_{[g]}\) by \({}^{[g]}\nabla\). This proves (4.10).

Suppose \(x=(x^i) : U \to \mathbb{R}^2\) are local orientation preserving \([g]\)-isothermal coordinates on \(\Sigma\) and write \(z=(x^1+\mathrm{i}x^2)\). Applying the exterior derivative to \(z\) we obtain a local section \(\tilde{z} : U \to F^+_{[g]}\) so that \[A_{[g]}=\tilde{z}^*A^i_{jk}\mathrm{d}x^j\otimes \mathrm{d}x^k\otimes \frac{\partial}{\partial x^i}.\] By definition of \(\alpha\) we have \[\alpha=\tilde{z}^*a\,\mathrm{d}z\otimes\mathrm{d}z\otimes \otimes \frac{\partial}{\partial \overline{z}},\] hence (4.11) is an immediate consequence of (4.21).

Finally, in our coordinates we obtain \[|A_{[g]}|_g^2\,d\mu_g=4|a|^2\mathrm{d}x^1\wedge \mathrm{d}x^2,\] so that \(p^*\left(|A_{[g]}|_g^2\,d\mu_g\right)=2\mathrm{i}|a|^2\zeta_1\wedge\overline{\zeta_1}=-\frac{\mathrm{i}}{2}\zeta_2\wedge\overline{\zeta_2}\), as claimed.

Proof. Let \([g] : \Sigma \to Z\) be a conformal structure and \([g]_t : \Sigma \to Z\) a smooth variation of \([g]\) with support in some compact set \(\Omega\subset \Sigma\) and with \(|t|<\varepsilon\). We consider the submanifold of \(\Sigma\times P\times (-\varepsilon,\varepsilon)\) defined by \[P^{\prime}_{[g]_t}=\left\{(p,u,t_0) \in \Sigma\times P\times (-\varepsilon,\varepsilon)\,|\,(p,u)\in P^{\prime}_{[g]_{t_0}} \right\}\] and denote by \(\iota_{[g]_t} : P^{\prime}_{[g]_t} \to \Sigma \times P\times (-\varepsilon,\varepsilon)\) the inclusion map. On \(\Sigma\times P\times (-\varepsilon,\varepsilon)\) we define the real-valued \(2\)-form \[\mathsf{A}=-\frac{\mathrm{i}}{2}\zeta_2\wedge\overline{\zeta_2},%.-\d t\wedge\left(\partial_t \inc\left(\zeta_2\wedge\ov{\zeta_2}\right)\right)\Bigg],\] where, by abuse of notation, we write \(\zeta_2\) for the pullback of \(\zeta_2\) to \(\Sigma\times P\times (-\varepsilon,\varepsilon)\). Using the structure equations (3.11), we compute \[\tag{4.22} \mathrm{d}\mathsf{A}=\frac{\mathrm{i}}{2}\left(\zeta_1\wedge\zeta_3\wedge\overline{\zeta_2}-\zeta_2\wedge\overline{\zeta_1}\wedge\overline{\zeta_3}\right).\]

Now Proposition 4.9 implies \[f(t_0):=\left.\mathcal{E}_{\mathfrak{p},\Omega}([g]_t)\right|_{t=t_0}=\int_{\Omega}\left.\left(\left(\iota_{[g]_t}\right)^*\mathsf{A}\right)\right|_{t=t_0}.\] Therefore \[f^{\prime}(0)=\int_{\Omega}\left.\left(\mathrm{L}_{\partial_t}(\iota_{[g]_t})^*\mathsf{A}\right)\right|_{t=0}=\int_{\Omega}\left.\left(\partial_t \hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}(\iota_{[g]_t})^*\mathrm{d}\mathsf{A}\right)\right|_{t=0},\] where \(\mathrm{L}_{\partial_t}\) denotes the Lie-derivative with respect to the vector field \(\partial_t\). It follows from the proof of Lemma 4.3 that on \(P^{\prime}_{[g]_t}\) there exist complex-valued functions \(a,k,q,B,C\) such that \[\tag{4.23} \zeta_2=2\overline{a}\overline{\zeta_1}+B\mathrm{d}t\quad \text{and}\quad \zeta_3=k\zeta_1+2\overline{q}\overline{\zeta_1}+C \mathrm{d}t\] where we now write \(\zeta_i\) instead of \((\iota_{[g]_t})^*\zeta_i\). Combining (4.22) with (4.23) gives \[(\iota_{[g]_t})^*\mathrm{d}\mathsf{A}=\mathrm{i}\left(qB+\overline{q} \overline{B}\right) \mathrm{d}t\wedge\zeta_1\wedge\overline{\zeta_1}\] so that \[\tag{4.24} f^{\prime}(0)=\mathrm{i}\int_{\Omega}\left.\left(qB+\overline{q}\overline{B}\right)\zeta_1\wedge\overline{\zeta_1}\right|_{t=0}.\] Recall that \(\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*\zeta_2=\mathrm{e}^{2\mathrm{i}\phi}\zeta_2\) and therefore, by definition, the complex-valued function \(B|_{t=0}\) satisfies \[\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*\left(B|_{t=0}\right)=\mathrm{e}^{2\mathrm{i}\phi}\left(B|_{t=0}\right).\] Since \(\left(R_{r\mathrm{e}^{\mathrm{i}\phi}}\right)^*\zeta_1=r^{-3}\mathrm{e}^{\mathrm{i}\phi}\zeta_1\) it follows that \(B|_{t=0}\) represents a section of \(\overline{K_{\Sigma}}\otimes K_{\Sigma}^{*}\) with support in \(\Omega\). Here \(K_{\Sigma}\) denotes the canonical bundle of \(\Sigma\) with respect to the complex structure induced by the orientation and \([g]=[g]_t|_{t=0}\).

It remains to show that every such section in (4.23) with support in \(\Omega\) can be realised via some variation of \([g]\). We fix a representative metric \(g \in [g]\). Let \(g_{ij}=g_{ji}\) be the real-valued functions on Cartan’s bundle \(P\) so that \(\pi^*g=g_{ij}\theta^i_0\otimes \theta^j_0\). In particular, from the equivariance properties (ii) of the Cartan connection \(\theta\) it follows that \[\left(R_{b\rtimes a}\right)^*\begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22}\end{pmatrix}=(\det a)^2a^t\begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22}\end{pmatrix}a.\] Applying property (iii) of the Cartan connection this implies the existence of unique real-valued functions \(g_{ijk}=g_{jik}\) so that \[\tag{4.25} \mathrm{d}g_{ij}=-2g_{ij}\theta^0_0+g_{kj}\theta^k_i+g_{ik}\theta^k_j+g_{ijk}\theta^k_0.\] Consider the following conformally invariant functions \[G=\frac{(g_{11}-g_{22})+2\mathrm{i}g_{12}}{\sqrt{g_{11}g_{22}-(g_{12})^2}}, \quad H=\frac{g_{11}+g_{22}}{\sqrt{g_{11}g_{22}-(g_{12})^2}}.\] Translating (4.25) into complex form gives the following structure equation \[\tag{4.26} \mathrm{d}G=G^{\prime}\zeta_1+G^{\prime\prime}\overline{\zeta_1}+H\zeta_2+\overline{G}\left(\overline{\varphi}-\varphi\right),\] for unique complex-valued functions \(G^{\prime},G^{\prime\prime}\) on \(P\). Clearly, the complex-valued functions \(G^{\prime}\) and \(G^{\prime\prime}\) can be expressed in terms of the functions \(g_{ijk}\), as \(\zeta_1=\theta^1_0+\mathrm{i}\theta^2_0\). In order to verify (4.26) it is thus sufficient to plug in the definitions of the functions \(G,H\), the definitions of the forms \(\zeta_2,\varphi\) and to use \[\mathrm{d}g_{ij}=-2g_{ij}\theta^0_0+g_{kj}\theta^k_i+g_{ik}\theta^k_j \quad \text{mod}\quad \theta^1_0,\theta^2_0.\] While this is somewhat tedious, it is straightforward, so we omit the computation.

Fix a section of \(\overline{K_{\Sigma}}\otimes K_{\Sigma}^{*}\) with respect to \([g]\) having support in \(\Omega\). Such sections are well-known to correspond to endomorphisms of \(T\Sigma\) that are trace-free and symmetric with respect to \([g]\). In particular, on \(P\) there exist real-valued functions \((B^i_j)\) representing the corresponding endomorphism. The functions satisfy \[B^i_i=0 \quad \text{and} \quad g_{ij}B^j_k=g_{kj}B^j_i.\] as well as the equivariance property \[\left(R_{b\rtimes a}\right)^*\begin{pmatrix} B^1_1 & B^1_2 \\ B^2_1 & B^2_2\end{pmatrix}=a^{-1}\begin{pmatrix} B^1_1 & B^1_2 \\ B^2_1 & B^2_2\end{pmatrix}a.\] We define \(B=\frac{1}{2}(B^1_1-B^2_2)+\frac{\mathrm{i}}{2}\left(B^1_2+B^2_1\right)\), then \(B\) satisfies \((R_{z\rtimes r\mathrm{e}^{\mathrm{i}\phi}})^*B=\mathrm{e}^{2\mathrm{i}\phi}B\), hence for sufficiently small \(t\) we may vary \([g]\) by defining \([g]_t\) via the zero-locus of the function \[G_t=G-tB H.\] Consequently, on \[P_{[g]_t}=\left\{(p,u,t_0)\in \Sigma\times P\times (-\varepsilon,\varepsilon)\,|\,(p,u) \in P_{[g]_{t_0}}\right\}\] we get \[\begin{aligned} 0&=\mathrm{d}G_t=\mathrm{d}G-\mathrm{d}tB H-t\mathrm{d}\left(B H\right)\\ &=G^{\prime}\zeta_1+G^{\prime\prime}\overline{\zeta_1}+H\zeta_2+\overline{G}\left(\overline{\varphi}-\varphi\right)-\mathrm{d}tB H-t\mathrm{d}\left(B H\right)\\ &=G^{\prime}\zeta_1+G^{\prime\prime}\overline{\zeta_1}+H\zeta_2+t\overline{B}H\left(\overline{\varphi}-\varphi\right)-\mathrm{d}tB H-t\mathrm{d}\left(B H\right) \end{aligned}\] In particular, if we evaluate this last equation on \(\left.P_{[g]_t}\right|_{t=0}\), we obtain \[0=G^{\prime}\zeta_1+G^{\prime\prime}\overline{\zeta_1}+H\zeta_2-\mathrm{d}t B H\] Since \(H\) is non-vanishing on \(\left.P_{[g]_t}\right|_{t=0}\) we must have \[\zeta_2=-\frac{G^{\prime}}{H}\zeta_1-\frac{G^{\prime\prime}}{H}\overline{\zeta_1}+B\mathrm{d}t.\] Since \(P_{[g]_t}^{\prime}\) arises by reducing \(P_{[g]_t}\), it follows that on \(\left.P^{\prime}_{[g]_t}\right|_{t=0}\) we obtain \[\zeta_2=-\frac{G^{\prime\prime}}{H}\overline{\zeta_1}+B \mathrm{d}t,\] as desired. Finally, we now know that (4.24) must vanish where \(B\) is any complex-valued function representing an arbitrary section of \(\overline{K_{\Sigma}}\otimes K_{\Sigma}^{*}\) with support in \(\Omega\). This is only possible if \(q|_{t=0}\) vanishes identically. Applying Corollary 4.6 proves the claim.

Proof. Recall from (4.9) that \[p^*\left(\widetilde{[g]}^*h_{\mathfrak{p}}\right)=\frac{1}{2}\left(4|a|^2+(k+\overline{k})\right)\zeta_1\circ\overline{\zeta_1}+q\,\zeta_1\circ\zeta_1+\overline{q}\,\overline{\zeta_1}\circ\overline{\zeta_1}.\] Hence we obtain \[\frac{1}{2}\int_{\Sigma}\operatorname{tr}_g \widetilde{[g]}^*h_{\mathfrak{p}}\, d\mu_g=\frac{1}{2}\int_{\Sigma}\left(4|a|^2+(k+\overline{k})\right)\frac{\mathrm{i}}{2}\zeta_1\wedge\overline{\zeta_1}.%=\int_{\Sigma}|A_{[g]}|^2_gd\mu_g\\ \] Since \[\mathrm{d}\varphi=\left(|a|^2+\frac{1}{2}k-\overline{k}\right)\zeta_1\wedge\overline{\zeta_1},\] we get \[\frac{\mathrm{i}}{2}\left(\mathrm{d}\varphi-\mathrm{d}\overline{\varphi}\right)=\frac{1}{2}\left(4|a|^2-(k+\overline{k})\right)\frac{\mathrm{i}}{2}\zeta_1\wedge\overline{\zeta_1}\] and thus \[\begin{aligned} \frac{1}{2}\int_{\Sigma}\operatorname{tr}_g \widetilde{[g]}^*h_{\mathfrak{p}}\, d\mu_g&=\int_{\Sigma}2\mathrm{i}|a|^2\zeta_1\wedge\overline{\zeta_1}-\int_{\Sigma}\frac{\mathrm{i}}{2}(\mathrm{d}\varphi-\mathrm{d}\overline{\varphi})\\ &=\int_{\Sigma}|A_{[g]}|^2_gd\mu_g-2\pi\chi(\Sigma), \end{aligned}\] where we have used (4.1) and (4.12).