4 Holomorphic curves

It is natural to ask whether for a given torsion-free connection \(\nabla\) on \(TM\) one can always (at least locally) choose a conformal structure \([g]\) on \(M\) so that \(\Phi\) vanishes identically. Equivalently, whether every torsion-free connection \(\nabla\) on \(TM\) is locally projectively equivalent to a Weyl connection \(\mathrm{D}\). This question was answered in the affirmative in [14], where it is also observed that the problem is equivalent to finding a suitable holomorphic curve into a complex surface fibering over \(M\). Here we will briefly review this observation and use it do derive a non-linear PDE for the Beltrami differential of the sought after conformal structure.

4.1 A complex surface

Inspired by the twistorial construction of holomorphic projective structures by Hitchin [8], it was shown in [5] and [19] and how to construct a ‘twistor space‘ for smooth projective structures. Let \(\nabla\) be a torsion-free connection on \(TM\) and \(\varphi=(\varphi^i_j) \in \Omega^1(P,\mathfrak{gl}(2,\mathbb{R}))\) its connection form on the frame bundle \(P\). We can use \(\varphi\) to construct a complex structure on the quotient \(P/\mathrm{CO}(2)\). By definition, an element of \(P/\mathrm{CO}(2)\) gives a frame in some tangent space of \(M\), well defined up to rotation and scaling. Therefore, the conformal structures on \(M\) are in one-to-one correspondence with the sections of the fibre bundle \(P/\mathrm{CO}(2) \to M\) whose fibre is \(\mathrm{GL}^+(2,\mathbb{R})/\mathrm{CO}(2)\), that is, the open disk. We will construct a complex structure on \(P/\mathrm{CO}(2)\) in terms of its \((1,\! 0)\)-forms, or more precisely, the pullbacks of the \((1,\! 0)\)-forms to \(P\). Recall that the Lie algebra \(\mathfrak{co}(2)\) of \(\mathrm{CO}(2)\) is spanned by the matrices \[\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\quad \text{and}\quad \begin{pmatrix} 0 & -1 \\ 1 & 0\end{pmatrix}.\] Consequently, the complex-valued \(1\)-forms on \(P\) that are semibasic for the quotient projection \(P \to P/\mathrm{CO}(2)\) are spanned by the form \(\omega\) and \[\zeta=(\varphi^1_1-\varphi^2_2)+i (\varphi^1_2+\varphi^2_1)\] as well as their complex conjugates. Recall that we have \(\left(R_{r\mathrm{e}^{i\phi}}\right)^*\omega=\frac{1}{r}\mathrm{e}^{-i\phi}\) and using that \(\varphi\) satisfies the equivariance property \(R_h^*\varphi=h^{-1}\varphi h\) for all \(h \in \mathrm{GL}^+(2,\mathbb{R})\), we compute \(\left(R_{r\mathrm{e}^{i\phi}}\right)^*\zeta=\mathrm{e}^{-2i\phi}\zeta\). It follows that there exists a unique almost complex structure \(J\) on \(P/\mathrm{CO}(2)\) whose \((1,\!0)\)-forms pull back to \(P\) to become linear combinations of the forms \(\omega,\zeta\). The almost complex structure \(J\) can be shown to only depend on the projective equivalence class of \(\nabla\) and moreover, an application of the Newlander–Nirenberg theorem shows that \(J\) is always integrable, see [14] for details.

4.2 Möbius action

In our setting it is convenient to reduce the frame bundle \(P\) to the unit tangent bundle \(SM\) of some fixed metric \(g\). In order to get a handle on the complex surface \(P/\mathrm{CO}(2)\) after having carried out this reduction, we interpret the disk bundle \(P/\mathrm{CO}(2) \to M\) as an associated bundle to the frame bundle \(P\). This requires an action of the structure group \(\mathrm{GL}^+(2,\mathbb{R})\) on the open disk and this is what we compute next.

The group \(\mathrm{GL}^+(2,\mathbb{R})\) acts from the left on the lower half plane \[-\mathbb{H}:=\left\{w \in {\mathbb C}: \Im(w)<0\right\}\] by Möbius transformations, where \(w\) denotes the standard coordinate on \({\mathbb C}\). We let \(\mathbb{D}\subset {\mathbb C}\) denote the open unit disk. Identifying \(-\mathbb{H}\) with \(\mathbb{D}\) via the Möbius transformation \[-\mathbb{H} \to \mathbb{D}, \quad w \mapsto -\left(\frac{w+i}{w-i}\right)\] we obtain an induced action of \(\mathrm{GL}^+(2,\mathbb{R})\) on \(\mathbb{D}\) making this transformation equivariant \[\tag{4.1} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot z=\frac{iz(a+d)+z(b-c)-i(a-d)+(b+c)}{-iz(a-d)-z(b+c)+i(a+d)-(b-c)}.\] The stabiliser subgroup of the point \(z=0\) consists of elements in \(\mathrm{GL}^+(2,\mathbb{R})\) satisfying \(a=d\) and \(b+c=0\), i.e., the linear conformal group \(\mathrm{CO}(2)\). Consequently, we have \(\mathbb{D}\simeq \mathrm{GL}^+(2,\mathbb{R})/\mathrm{CO}(2)\) and we obtain a projection \[\lambda : \mathrm{GL}^+(2,\mathbb{R}) \to \mathbb{D}, \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot 0=\frac{-i(a-d)+(b+c)}{i(a+d)-(b-c)}.\] In particular, a mapping \(z : N \to \mathbb{D}\) from a smooth manifold \(N\) into \(\mathbb{D}\) is covered by a map \[\tilde{z}=\begin{pmatrix} \frac{1-|z|^2}{(1+z)(1+\overline z)} & \frac{i(z-\overline{z})}{(1+z)(1+\overline z)} \\ 0 & 1\end{pmatrix}.\] into \(\mathrm{GL}^+(2,\mathbb{R})\). Equivalently, we have \(\tilde{z}\cdot 0=z\) or \(z \cdot \tilde{z}=0\), where as usual we turn the left action into a right action by the definition \(z\cdot \tilde{z}:=\tilde{z}^{-1}\cdot z\).

Let \(\rho : Z \to M\) denote the disk-bundle associated to the above \(\mathrm{GL}^+(2,\mathbb{R})\) action on \(\mathbb{D}\). Suppose \(z : P \to \mathbb{D}\) represents a section of \(Z \to M\) so that \(z\) is a \(\mathrm{GL}^+(2,\mathbb{R})\)-equivariant map. For every coframe \(u \in P\) the pair \((u,z(u)) \in P \times \mathbb{D}\) lies in the same \(\mathrm{GL}^+(2,\mathbb{R})\) orbit as \[\tag{4.2} (u\cdot \tilde{z}(u),z(u)\cdot\tilde{z}(u))=(u\cdot \tilde{z}(u),0).\] Therefore, the map \(z\) gives for every point \(p \in M\) a coframe \(u\cdot \tilde{z}(u)\) which is unique up to the action of \(\mathrm{CO}(2)\). It follows that the bundle \(Z \to M\) is isomorphic to \(P/\mathrm{CO}(2) \to M\), as desired.

Let \(\Upsilon : P\times \mathbb{D} \to P\) be the map defined by (4.2). We will next compute the pullback of \(\omega,\zeta\) under \(\Upsilon\). Note that we may write \(\Upsilon=R\circ\left(\mathrm{Id}_P\times \tilde{z}\right)\) where \(R : P \times \mathrm{GL}^+(2,\mathbb{R}) \to P\) denotes the \(\mathrm{GL}^+(2,\mathbb{R})\) right action of \(P\). Recall the standard identities \[R^*\varphi=h^{-1}\varphi h+ h^{-1}dh\quad \text{and} \quad R^*\omega=h^{-1}\omega,\] where \(h : P\times \mathrm{GL}^+(2,\mathbb{R}) \to \mathrm{GL}^+(2,\mathbb{R})\) denotes the projection onto the latter factor. From this we compute \[\tag{4.3} \omega_\Upsilon:=\Upsilon^*\omega=\tilde{z}^{-1}\omega=\left(\frac{1+\overline{z}}{1-|z|^2}\right)\left(\omega+z\overline{\omega}\right).\] and \[\tag{4.4} \varphi_{\Upsilon}:=\Upsilon^*\varphi=\tilde{z}^{-1}\varphi \tilde{z}+\tilde{z}^{-1}d\tilde{z}\] We also obtain \(\zeta_{\Upsilon}=\Upsilon^*\zeta=(\varphi_{\Upsilon})^1_1-(\varphi_{\Upsilon})^2_2+i\left((\varphi_{\Upsilon})^1_2+(\varphi_{\Upsilon})^2_1\right)\). Writing \[\chi=\frac{1}{2}\left(3(\varphi^1_1+\varphi^2_2)+i(\varphi^2_1-\varphi^1_2)\right),\] and using (4.4), a tedious but straightforward calculation gives \[\tag{4.5} \zeta_\Upsilon=\frac{2(1+\overline{z})}{(|z|^2-1)(z+1)}\left(dz-\frac{1}{2}\zeta+\frac{1}{2}z^2\overline{\zeta}+z\chi-z\bar{\chi}\right).\]

The complex structure on \(Z\) does only depend on the projective equivalence class of \(\nabla\). Thus, after possibly replacing \(\varphi\) with a projectively equivalent connection, we can assume that the torsion-free connection on \(TM\) corresponding to \(\varphi\) is of the form \(\mathrm{D}+B\) for some \(1\)-form \(\theta\) and some cubic differential \(A\) on \(M\). On the unit tangent bundle \(SM\) of \(g\) the connection form of \(\mathrm{D}+B\) takes the form (3.3). Using this equation and reducing to \(SM\subset P\) yields the following identities on \(SM\) \[\tag{4.6} \begin{align} \zeta&=2a_{-3}\overline{\omega}, \\ \chi&=i\psi-4\theta_1\omega-2\theta_{-1}\overline{\omega}, \end{align}\] Recall, we write \(a_3=\frac{1}{3}Va+ia\) and \(a_{-3}=\overline{a_3}\) as well as \(\theta_1=\frac{1}{2}(\theta-iV\theta)\) and \(\theta_{-1}=\overline{\theta_1}\). Also, the connection form \(\kappa\) of the induced Weyl connection is \(\kappa=i\psi-2\theta_1\omega\), see (2.8). Therefore, we have \[\chi=2\kappa+\overline{\kappa}.\]

The \(\mathrm{SO}(2)\)-action induced by (4.1) is \[\begin{pmatrix}\cos \phi & -\sin\phi \\ \sin\phi & \cos\phi\end{pmatrix}\cdot z=\frac{2iz\cos\phi-2z\sin\phi}{2i\cos\phi+2\sin\phi}=\mathrm{e}^{2i\phi}z\] and hence the equivariance property of a function \(z : SM \to \mathbb{D}\) representing a section of \(Z \to M\) becomes \(\left(R_{\mathrm{e}^{i\phi}}\right)^*z=\mathrm{e}^{-2i\phi}z\), that is, \(z\) represents a section of \(K^{-2}\). Since we have a metric, we have an identification \(K^*\simeq \overline{K}\) and hence \(K^{-2}\simeq K^*\otimes \overline{K}\). In particular, we may write \[\tag{4.7} dz=z^{\prime}\omega+z^{\prime\prime}\overline\omega+\overline{\kappa}z-\kappa z\] For unique complex-valued functions \(z^{\prime}\) and \(z^{\prime\prime}\) on \(SM\). Consequently, using (4.5), (4.6) and (4.7) we obtain \[\tag{4.8} \begin{align} \left(\frac{(|z|^2-1)(z+1)}{2(1+\overline{z})}\right)\zeta_\Upsilon&=z^{\prime}\omega+z^{\prime\prime}\overline{\omega}+\overline{\kappa}z-\kappa z-a_{-3}\overline{\omega}+z^2a_3\omega\\ &\phantom{=}+z(2\kappa+\overline{\kappa})-z(2\overline{\kappa}+\kappa)\\ &=\left(z^{\prime}+z^2a_3\right)\omega+\left(z^{\prime\prime}-a_{-3}\right)\overline{\omega}. \end{align}\] In order to connect the expressions for \(\omega_\Upsilon\) and \(\zeta_\Upsilon\) to the condition of \(z\) representing a conformal structure \([\hat{g}]\) that defines a holomorphic curve into \(Z\), we use the following elementary lemma:

The reduction of \(P\) to \(SM\) identifies \(Z\) with \(SM\times_{\mathrm{SO}(2)} \mathbb{D}\). Now suppose the conformal structure \([\hat{g}] : M \to Z\) is represented by the map \(z : SM \to \mathbb{D}\). If \(v : U \to SM\) is a local section of \(\pi : SM \to M\), then \([\hat{g}]|_U : U \to Z\) is covered by the map \((\mathrm{Id}_{SM}\times z)\circ v : U \to SM \times \mathbb{D}\). Recall that the complex structure on \(Z\) has the property that its \((1,\! 0)\)-forms pull-back to become linear combination of \(\omega_\Upsilon\) and \(\zeta_\Upsilon\). Using the expressions (4.3) and (4.8) for the pullbacks of \(\omega_\Upsilon\) and \(\zeta_\Upsilon\) to \(SM\) we obtain \[\omega_\Upsilon\wedge\zeta_\Upsilon=-\frac{2(1+\overline{z})^2}{(|z|^2-1)^2(z+1)}\left(z^{\prime\prime}-zz^{\prime}-z^3a_3-a_{-3}\right)\omega\wedge\overline{\omega}.\] In particular, since \(v : U \to SM\) is a \(\pi\)-section and \(\omega\) and \(\overline{\omega}\) are \(\pi\)-semibasic, the pullback \(v^*(\omega_\Upsilon\wedge\zeta_\Upsilon)\) vanishes if and only if \(\omega_\Upsilon\wedge\zeta_\Upsilon\) vanishes on \(\pi^{-1}(U)\). Thus, Lemma 4.3 implies that \(z\) represents a holomorphic curve if and only if \[\tag{4.9} z^{\prime\prime}-zz^{\prime}=z^3a_3+a_{-3}.\]

4.3 The Beltrami differential

So far we have not explicitly tied the conformal structure \([\hat{g}]\) to the function \(z : SM \to \mathbb{D}\) representing it. In order to do this we first recall the Beltrami differential. The choice of a metric \(\hat{g}\) on \(M\) allows to define the functions \[p(x,v)=\hat{g}(v,v), \qquad r(x,v)=\hat{g}(v,Jv) \quad\text{and}\quad q(x,v)=\hat{g}(Jv,Jv)\] on \(SM\). The orientation compatible complex structure \(\hat{J}\) on \(M\) induced by the conformal equivalence class of \(\hat{g}\) has matrix representation \[\hat{J}=\frac{1}{\sqrt{pq-r^2}}\begin{pmatrix} -r & -q \\ p & r\end{pmatrix}.\] In particular, we compute that the \((1,\! 0)\)-forms with respect to \(\hat{J}\) pull-back to \(SM\) to become complex multiples of \[\tag{4.10} \omega_{\hat{J}}:=\frac{1}{2}\left(\omega-i\hat{J}\omega\right)=\left(\frac{p+q+2\sqrt{pq-r^2}}{4\sqrt{pq-r^2}}\right)\left(\omega+\mu\overline{\omega}\right)\] where \[\mu=\frac{(p-q)+2ir}{p+q+2\sqrt{pq-r^2}}\] is the Beltrami coefficient of \(\hat{J}\). Clearly, \(\mu\) does only depend on the conformal equivalence class \([\hat{g}]\) of \(\hat{g}\). Moreover, the function \(\mu\) represents a \((0,\! 1)\)-form on \(M\) with values in \(K^*\) called the Beltrami differential of \([\hat{g}]\), which – by abuse of language – we denote by \(\mu\) as well.

The reduction of \(P\) to the unit tangent bundle \(SM\) of \(g\) turned \(\omega\) into a basis for the \((1,\! 0)\)-forms with respect to the complex structure induced by \(g\) and the orientation. The mapping \(z\) represents a conformal structure \([\hat{g}]\) and consequently, induces an orientation compatible complex structure \(\hat{J}\) whose \((1,\! 0)\)-forms we computed in (4.3). Comparing this expression with the formula (4.10) for the Beltrami coefficient shows that we obtain the same \((1,\! 0)\)-forms if and only if \(z=\mu\). Remember, \(z^{\prime}\) and \(z^{\prime\prime}\) represent the \((1,\! 0)\) and \((0,\! 1)\) part of the derivative of \(z\) with respect to the connection \(\mathrm{D}\) induced by the Weyl connection \(\mathrm{D}\). Furthermore, the function \(a_{3}\) represents the cubic differential \(A\) or equivalently, the form \(\Phi\), since \(\Phi d\sigma=A\) and \(d\sigma\) is represented by the constant function \(1\) on \(SM\). Using equation (4.9) and the fact that \(\mathfrak{p}\) contains a Weyl connection with respect to \([\hat{g}]\) if and only if \([\hat{g}] : M \to Z\) is a holomorphic curve [14], we have thus shown:

As a corollary, we obtain: