5 Existence of critical points
Clearly, if a projective structure \(\mathfrak{p}\) is defined by a \([g]\)-conformal connection, then the conformal structure \([g]\) is a critical point for \(\mathcal{E}_\mathfrak{p}\) and moreover an absolute minimiser. In this final section we study the projective structures for which \(\mathcal{E}_{\mathfrak{p}}\) admits a critical point in some more detail. In particular, we will prove that properly convex projective structures admit critical points.
Recall that the choice of a conformal structure \([g]\) on an oriented projective surface \((\Sigma,\mathfrak{p})\) determines a torsion-free principal \(\mathrm{CO}(2)\)-connection \(\varphi\) on the bundle \(F^+_{[g]}\) of complex linear coframes of \((\Sigma,[g])\) and a section \(\alpha\) of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\). Furthermore, the conformal structure \([g]\) is extremal for \(\mathcal{E}_{\mathfrak{p}}\) if and only if \(\nabla^{\prime\prime}_\varphi \alpha=0\). Conversely, let \((\Sigma,[g])\) be a Riemann surface. Let \(\varphi\) be a torsion-free principal \(\mathrm{CO}(2)\)-connection on \(F^+_{[g]}\) and \(\alpha\) a section of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\). Then Proposition 4.5, Proposition 4.9 and Theorem A show that the conformal structure \([g]\) is extremal for the projective structure defined by \(\nabla_{\varphi}+2\operatorname{Re}(\alpha)\) if and only if \(\nabla_{\varphi}^{\prime\prime}\alpha\equiv 0\). Since the curvature of the connection induced by \(\varphi\) on the complex line bundle \(E=K^2_{\Sigma}\otimes\overline{K^*_{\Sigma}}\) is a \((1,\! 1)\)-form, standard results imply (see for instance [25]) that there exists a unique holomorphic line bundle structure \(\overline{\partial}_E\) on \(E\), so that \[\overline{\partial}_E=\nabla^{\prime\prime}_\varphi.\] Hence the variational equation \(\nabla^{\prime\prime}_\varphi\alpha=0\) just says that \(\alpha\) is holomorphic with respect to \(\overline{\partial}_E\). Since the line bundle \(E\) has degree \[\deg(E)=\deg(K^2_{\Sigma})-\deg\left(K^*_{\Sigma}\right)=-3\deg\left(K^*_\Sigma\right)=-3\chi(\Sigma),\] we immediately obtain:
Suppose \(\mathfrak{p}\) is a projective structure on the oriented \(2\)-sphere \(S^2\) admitting an extremal conformal structure \([g]\). Then \(\mathfrak{p}\) is defined by a \([g]\)-conformal connection.
Proof. Suppose \([g]\) is an extremal conformal structure of \(\mathcal{E}_{\mathfrak{p}}\). From Corollary 2.6 we know that \(\mathfrak{p}\) is defined by \({}^{[g]}\nabla+A_{[g]}\) for some \([g]\)-conformal connection \({}^{[g]}\nabla\). Since \(\chi(S^2)=2\), we have \(\deg(E)=-6\) and hence the only holomorphic section of \(E\) is the zero-section. It follows that \(\alpha\) vanishes identically and since by Proposition 4.9 we have \(A_{[g]}=2\operatorname{Re}(\alpha)\), so does \(A_{[g]}\).
Note that the projectively flat conformal connections on \(S^2\) are classified in [37].
From the Riemann–Roch theorem we know that the space \(H^0(\Sigma,E)\) of holomorphic sections of \(E\) has dimension \[\dim_{\mathbb{C}}\,H^0(\Sigma,E)\geqslant \deg(E)+1-g_{\Sigma}=5g_{\Sigma}-5,\] where here \(g_{\Sigma}\) denotes the genus of \(\Sigma\). In particular, if \(\Sigma\) has negative Euler-characteristic, then \(\dim_{\mathbb{C}} H^0(\Sigma,E)\) will have positive dimension.
5.1 Convex projective structures
Recall that a flat projective surface \((\Sigma,\mathfrak{p})\) has the property that \(\Sigma\) can be covered with open subsets, each of which is diffeomorphic onto a subset of \(\mathbb{RP}^2\) in such a way that the geodesics of \(\mathfrak{p}\) are mapped onto (segments) of projective lines \(\mathbb{RP}^1\subset \mathbb{RP}^2\). This condition turns out to be equivalent to \(\Sigma\) carrying an atlas modelled on \(\mathbb{RP}^2\), that is, an atlas whose chart transitions are restrictions of fractional linear transformations. On the universal cover \(\tilde{\Sigma}\) of the surface the charts can be adjusted to agree on overlaps, thus defining a developing map \(\mathrm{dev} : \tilde{\Sigma} \to \mathbb{RP}^2\), unique up to post-composition with an element of \(\mathrm{SL}(3,\mathbb{R})\). In addition, one obtains a monodromy representation \(\rho :\pi_1(\Sigma) \to \mathrm{SL}(3,\mathbb{R})\) of the fundamental group \(\pi_1(\Sigma)\) – well defined up to conjugation – making \(\mathrm{dev}\) into an equivariant map. A flat projective structure is called properly convex if \(\mathrm{dev}\) is a diffeomorphism onto a subset of \(\mathbb{RP}^2\) which is bounded and convex. If \(\Sigma\) is a compact orientable surface with negative Euler characteristic, then (the conjugacy class of) ‘the’ monodromy representation \(\rho\) of a properly convex projective structure is an element in the Hitchin component \(\mathcal{H}_3\) of \(\Sigma\) and conversely every element in \(\mathcal{H}_3\) can be obtained in this way [10].
Motivated by the circle of ideas discussed in the introduction, it is shown in [28] and [34] that on a compact oriented surface \(\Sigma\) of negative Euler characterstic, the convex projective structures are parametrised in terms of pairs \(([g],C)\), consisting of a conformal structure \([g]\) and a cubic differential \(C\) that is holomorphic with respect to the complex structure induced by \([g]\) and the orientation. Indeed, given a holomorphic cubic differential \(C\) on such a \(\Sigma\), there exists a unique Riemannian metric \(g\) in the conformal equivalence class \([g]\), so that \[\tag{5.1} K_g=-1+2|C|^2_g,\] where \(K_g\) denotes the Gauss curvature of \(g\) and \(|C|_g\) the pointwise norm of \(C\) with respect to the Hermitian metric induced by \(g\) on the third power of the canonical bundle \(K_{\Sigma}\) of \(\Sigma\). Now there exists a unique section \(\alpha\) of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\), so that \(\alpha\otimes d\mu_g=C\), where here we think of the area form \(d\mu_g\) of \(g\) as a section of \(K_{\Sigma}\otimes \overline{K_{\Sigma}}\). Consequently, we obtain a connection \(\nabla={}^{g}\nabla+2\operatorname{Re}(\alpha)\) on \(T\Sigma\). The projective structure defined by \(\nabla\) is properly convex and conversely every properly convex projective structure arises in this way [28]. The metric \(g\) is known as the affine metric or Blaschke metric, due to the fact that its pullback to the universal cover \(\tilde{\Sigma}\) of \(\Sigma\) can be realised via some immersion \(\tilde{\Sigma} \to \mathbb{A}^3\) as a complete hyperbolic affine \(2\)-sphere in the affine \(3\)-space \(\mathbb{A}^3\). In particular, (5.1) is known as Wang’s equations in the affine sphere literature [43]. We refer the reader to the survey articles [23], [33] as well as [1] for additional details.
Calling a conformal structure \([g]\) on \((\Sigma,\mathfrak{p})\) closed, if the associated connection \(\varphi\) on \(F^+_{[g]}\) induces a flat connection on \(\Lambda^2(T^*\Sigma)\), we obtain a novel characterisation of properly convex projective structures among flat projective structures:
Let \((\Sigma,\mathfrak{p})\) be a compact oriented flat projective surface of negative Euler characteristic. Suppose \(\mathfrak{p}\) is properly convex, then the conformal equivalence class of the Blaschke metric is closed and extremal for \(\mathcal{E}_{\mathfrak{p}}\). Conversely, if \(\mathcal{E}_{\mathfrak{p}}\) admits a closed extremal conformal structure \([g]\), then \(\mathfrak{p}\) is properly convex and \([g]\) is the conformal equivalence class of the Blaschke metric of \(\mathfrak{p}\).
It would be interesting to know if flat projective surfaces \((\Sigma,\mathfrak{p})\) exist for which \(\mathcal{E}_{\mathfrak{p}}\) admits an extremal conformal structure that is not closed.
Proof of Theorem C. Assume \(\mathfrak{p}\) is properly convex and let \(([g],C)\) be the associated pair. Let \(g\) the Blaschke metric satisfying (5.1) and \(\varphi\) the connection on \(F^+_{[g]}\) induced by the Levi-Civita connection of \(g\). Recall that \(\nabla_{\varphi}\) denotes the connection induced by \(\varphi\) on \(TM\), hence here we have \(\nabla_{\varphi}={}^g\nabla\). From [28] we know that \(\mathfrak{p}\) is defined by a connection of the form \[\nabla={}^g\nabla+2\operatorname{Re}(\alpha),\] where \(\alpha\) satisfies \(\alpha\otimes d \mu_g=C\). A simple computation shows that a torsion-free connection \(\varphi\) on \(F^+_{[g]}\) induces a flat connection on \(\Lambda^2(T^*\Sigma)\) if and only if \(\nabla_{\varphi}\) has symmetric Ricci tensor. Since here \(\nabla_{\varphi}={}^g\nabla\) is a Levi-Civita connection, it follows that the conformal structure defined by the Blaschke metric is closed. In addition, since \(C\) is holomorphic, we have \(\nabla_{\varphi}^{\prime\prime} C=0\) and furthermore, since \(d\mu_g\) is parallel with respect to \({}^{g}\nabla\), it follows that \(\nabla_{\varphi}^{\prime\prime} \alpha\) vanishes identically, thus showing that the conformal structure defined by the Blaschke metric is extremal for \(\mathcal{E}_{\mathfrak{p}}\).
Conversely, let \((\Sigma,\mathfrak{p})\) be a compact oriented flat projective surface of negativ Euler characteristic. Suppose \([g]\) is a closed and extremal conformal structure for \(\mathfrak{p}\). We let \(\varphi\) denote the induced connection on \(F^+_{[g]}\) and \(\alpha\) the corresponding section of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\). Lemma 4.4 implies that on \(P^{\prime}_{[g]}\simeq F^+_{[g]}\) we have the following structure equations, where we write \(\omega\) instead of \(\zeta_1\) \[\tag{4.7} \begin{aligned} \mathrm{d}a&=a^{\prime}\omega-q \overline{\omega}+2a\varphi-a\overline{\varphi},\\ \mathrm{d}k&=k^{\prime}\omega+k^{\prime\prime}\overline{\omega}+k\varphi+k\overline{\varphi},\\ \mathrm{d}q&=q^{\prime}\omega+\frac{1}{2}\left(\overline{L}+\overline{k^{\prime\prime}}-2\overline{q}a\right)\overline{\omega}+2q \varphi,\\ \mathrm{d}\varphi&=\left(|a|^2+\frac{1}{2}k-\overline{k}\right)\omega\wedge\overline{\omega}. \end{aligned}\] Since \([g]\) is extremal, we know that \(Q\) and hence \(q\) vanishes identically. Moreover, recall that \(\mathfrak{p}\) is flat if and only if \(L\equiv 0\), hence the third structure equation gives \[0=\mathrm{d}q=q^{\prime}\omega+\frac{1}{2}\overline{k^{\prime\prime}}\overline{\omega}\] showing that the functions \(q^{\prime}\) and \(k^{\prime\prime}\) vanish identically as well. Lemma 4.1 implies that \(-(\varphi+\overline{\varphi})\) is the connection form of the connection induced by \(\varphi\) on \(\Lambda^2(T^*\Sigma)\). Since \([g]\) is closed, the induced connection is flat and hence \(\mathrm{d}(\varphi+\overline{\varphi})\) must vanish identically. Thus we obtain \[0=\mathrm{d}(\varphi+\overline{\varphi})=\frac{3}{2}\left(\overline{k}-k\right)\omega\wedge\overline{\omega},\] showing that \(k\) must be real-valued. Note that since \(k\) is real-valued, we have \[0=\mathrm{d}(k-\overline{k})=k^{\prime}\omega-\overline{k^{\prime}}\overline{\omega},\] so that \(k^{\prime}\) vanishes identically. Finally, we have reduced the structure equations to \[\tag{5.3} \begin{aligned} \mathrm{d}a&=a^{\prime}\omega+2a\varphi-a\overline{\varphi},\\ \mathrm{d}k&=k\varphi+k\overline{\varphi},\\ \mathrm{d}\varphi&=\left(|a|^2-\frac{1}{2}k\right)\omega\wedge\overline{\omega}. \end{aligned}\] The equivariance property of the tautological \(1\)-form \(\omega\) on \(F^+_{[g]}\) gives \[(R_{r\mathrm{e}^{\mathrm{i}\phi}})^*\omega=\frac{1}{r}\mathrm{e}^{\mathrm{i}\phi}\omega\] for all \(r \mathrm{e}^{\mathrm{i}\phi} \in \mathrm{CO}(2)\). The function \(k\) represents a \((1,\! 1)\)-form \(\kappa\) on \(\Sigma\) which satisfies \(\upsilon^*\kappa=\frac{\mathrm{i}}{2}k\omega\wedge\overline{\omega}\). Consequently, \(k\) has the equivariance property \((R_{r\mathrm{e}^{\mathrm{i}\phi}})^*k=r^2k\). Recall that \[\int_{\Sigma} \mathrm{i}\mathrm{d}\varphi=\int_{\Sigma}\mathrm{i}\left(|a|^2-\frac{1}{2}k\right)\omega\wedge\overline{\omega}= 2\pi\chi(\Sigma)<0,\] hence \(k\) must be positive somewhere. Note that (5.3) shows that the \((1,\! 1)\)-form \(\kappa\) represented by \(k\) is parallel with respect to \(\varphi\). Consequently, \(k\) cannot vanish. Since \(\Sigma\) is assumed to be connected, the equivariance property of \(k\) implies that the equation \(k=1\) defines a reduction \(F^+_g \subset F^+_{[g]}\) to an \(\mathrm{SO}(2)\)-subbundle which is the orthonormal coframe bundle of a unique representative metric \(g \in [g]\). On \(F^+_g\) we have \[0=\mathrm{d}k=\varphi+\overline{\varphi},\] showing that we may write \(\varphi=\mathrm{i}\phi\) for a unique \(1\)-form \(\phi\) on \(F^+_g\). Of course, \(\phi\) is the Levi-Civita connection form of \(g\) and hence using \(\omega=\omega^1+\mathrm{i}\omega^2\), we obtain the familiar structure equation for the Levi-Civita connection of an oriented Riemannian \(2\)-manifold \[\mathrm{d}\phi=-\left(-1+2|a|^2\right)\omega^1\wedge\omega^2.\] We may define a cubic differential \(C\) by setting \(C=\alpha\otimes d\mu_g\) and since the pullback to \(F^+_g\) of the area form of \(g\) is \(\omega^1\wedge\omega^2\), we conclude the the cubic differential \(C\) is holomorphic and represented by the function \(a\). Since \[\mathrm{d}\phi=-K_g\omega^1\wedge\omega^2,\] where \(K_g\) denotes the Gauss curvature of \(g\), we have \[K_g=-1+2|C|^2_g,\] where we use that \(\upsilon^*|C|^2_g=|c|^2\). It follows that \(g\) is the Blaschke metric associated to the pair \(([g],C)\) and hence \(\mathfrak{p}\) is a properly convex projective structure.
5.2 Concluding remarks
Let \(G_0\) be a real split simple Lie group and \(S(G_0)\) the associated symmetric space. For our purposes we may take \(G_0=\mathrm{SL}(3,\mathbb{R})\) so that \(S(G_0)=\mathrm{SL}(3,\mathbb{R})/\mathrm{SO}(3)\), but the following results hold in the more general case. Suppose \(\Sigma\) is a compact oriented surface of negative Euler characteristic and \(\rho : \pi_1(\Sigma) \to G_0\) a representation in the Hitchin component for \(G_0\). By a theorem of Corlette [11], the choice of a conformal structure \([g]\) on \(\Sigma\) determines a map \(\psi : \tilde{\Sigma} \to S(G_0)\) which is equivariant with respect to \(\rho\) and harmonic with respect to the Riemannian metric on \(S(G_0)\) and the conformal structure on \(\tilde{\Sigma}\) obtained by lifting \([g]\). Furthermore, this map is unique up to post-composition with an isometry of \(S(G_0)\). The energy density of the map \(\psi\) descends to define a \(2\)-form \(e_{\rho}([g])\,d\mu_g\) on \(\Sigma\) and hence one may define an energy functional [12], [29] \[\mathcal{E}_{\rho}([g])=\int_{\Sigma} e_{\rho}([g])\,d\mu_g.\] The energy \(\mathcal{E}_{\rho}([g])\) turns out to only depend on the diffeotopy class of \([g]\) and thus defines an energy functional on Teichmüller space for every representation \(\rho\) in the Hitchin component of \(G_0\). The Hopf differential of the map \(\psi\) yields a holomorphic quadratic differential which descends to \(\Sigma\) as well and it is conjectured [17], [29], that for every representation in the Hitchin component there exists a unique conformal structure on \(\Sigma\) whose associated Hopf differential vanishes identically. For such a conformal structure the mapping \(\psi\) is harmonic and conformal, hence minimal. In [30] Labourie proves the existence of a unique \(\rho\)-equivariant minimal mapping \(\psi : \tilde{\Sigma} \to S(G_0)\) in the case where \(G_0\) has rank two (the case \(G_0=\mathrm{SL}(3,\mathbb{R})\) was treated previously in [28]). Labourie also shows the existence of such a mapping without any assumption on the rank of \(G_0\) in [29]. Moreover, in [30], the energy bound \[\mathcal{E}_{\rho}([g])\geqslant -2\pi \chi(\Sigma)\] is obtained, with equality if and only if \(\rho\) is a Fuchsian representation.
Given our results it is natural to expect a relation between \(\mathcal{E}_{\rho}\) and our functional \(\mathcal{E}_{\mathfrak{p}}\), where \(\rho\) is an element in the \(\mathrm{SL}(3,\mathbb{R})\) Hitchin component and \(\mathfrak{p}\) denotes its associated properly convex projective structure. However, relating the representation \(\rho\) to its associated projective structure \(\mathfrak{p}\) in a way that would allow to establish the expected relation proves to be quite difficult. This may be investigated elsewhere.
Although we are currently unable to prove this, the previous remark suggests that in the case of a properly convex compact oriented projective surface \((\Sigma,\mathfrak{p})\) of negative Euler characteristic, the conformal equivalence class of the Blaschke metric is in fact the unique critical point of \(\mathcal{E}_{\mathfrak{p}}\). As a partial result towards this claim, it is shown in [41] that if a properly convex compact oriented projective surface \((\Sigma,\mathfrak{p})\) of negative Euler characteristic admits a compatible Weyl connection, then \(\mathfrak{p}\) arises from a hyperbolic metric.
In [38], it is shown that for a compact oriented projective surface \((\Sigma,\mathfrak{p})\) of negative Euler characteristic the functional \(\mathcal{E}_{\mathfrak{p}}\) admits at most one absolute minimiser \([g]\) (i.e. a conformal structure \([g]\) such that \(\mathcal{E}_{\mathfrak{p}}([g])=0\)).
In [39], the author shows that properly convex projective surfaces arise from torsion-free connections on \(T\Sigma\) that admit an interpretation as Lagrangian minimal surfaces. Some of their properties are studied in [40]. It would be interesting to relate these minimal Lagrangian surfaces to the minimal mapping \(\psi\) constructed in [28].
We have seen that oriented projective structures admitting extremal conformal structures arise from pairs \((\varphi,\alpha)\) on a Riemann surface \((\Sigma,[g])\), where \(\alpha\) satisfies \(\nabla^{\prime\prime}_{\varphi}\alpha\equiv 0\). The torsion-free connection \(\varphi\) on \(F^+_{[g]}\) induces a holomorphic line bundle structure \(\overline{\partial}_E\) on \(E=K_{\Sigma}^2\otimes \overline{K^*_{\Sigma}}\) and conversely, it is easy see that for every choice of a holomorphic line bundle structure \(\overline{\partial}_E\) on \(E\) there exists a unique torsion-free connection \(\varphi\) on \(F^+_{[g]}\) inducing \(\overline{\partial}_E\). Hence we may equivalently describe these projective structures in terms of a pair \((\overline{\partial}_E,\alpha)\) satisfying \(\overline{\partial}_E\alpha\equiv 0\).
The so-called naive Einstein affine hypersurface structures introduced in [16] also provide examples of projective surfaces admitting an extremal conformal structure.