1 Introduction

A projective structure on an \(n\)-manifold \(M\) is an equivalence class \(\mathfrak{p}\) of torsion-free connections on the tangent bundle \(TM\), where two connections are called projectively equivalent if they share the same unparametrised geodesics. A manifold \(M\) equipped with a projective structure \(\mathfrak{p}\) will be called a projective manifold. A conformal structure on \(M\) is an equivalence class \([g]\) of Riemannian metrics on \(M\), where two metrics are called conformally equivalent if they differ by a scale factor. Naively, one might think of projective and conformal structures as formally similar, since both arise by defining a notion of equivalence on a geometric structure. However, the formal similarity is more substantial. For instance, Kobayashi has shown [24] that both projective – and conformal structures admit a treatment as Cartan geometries with \(|1|\)-graded Lie algebras. Here we exploit the fact that both structures give rise to affine subspaces modelled on \(\Omega^1(M)\) of the infinite-dimensional affine space \(\mathfrak{A}(M)\) of torsion-free connections on \(M\). Indeed, it is a classical result due to Weyl [44] that two torsion-free connections on \(TM\) are projectively equivalent if and only if their difference – thought of as a section of \(S^2(T^*M)\otimes TM\) – is pure trace. Consequently, the representative connections of a projective structure \(\mathfrak{p}\) on \(M\) define an affine subspace \(\mathfrak{A}_{\mathfrak{p}}(M)\) which is modelled on \(\Omega^1(M)\). Moreover, it follows from Koszul’s identity, that the torsion-free connections preserving a conformal structure \([g]\) on \(M\) are of the form \[{}^g\nabla+g\otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes\beta,\] with \(g \in [g]\), \(\beta \in \Omega^1(M)\) and where \({}^g\nabla\) denotes the Levi-Civita connection of \(g\). Hence, the space of torsion-free \([g]\)-conformal connections on \(TM\) is an affine subspace \(\mathfrak{A}_{[g]}(M)\) modelled on \(\Omega^1(M)\) as well. It is an elementary computation to check that if \(\mathfrak{A}_{[g]}(M)\) and \(\mathfrak{A}_{\mathfrak{p}}(M)\) intersect, then they do so in a unique point. Therefore, we may ask if in general one can distinguish a point in \(\mathfrak{A}_{\mathfrak{p}}(M)\) and a point in \(\mathfrak{A}_{[g]}(M)\) which are ‘as close as possible’. This is indeed the case. More precisely, we show that the choice of a conformal structure \([g]\) on \((M,\mathfrak{p})\) determines a \(1\)-form \(A_{[g]}\) on \(M\) with values in the endomorphisms of \(TM\), as well as a unique \([g]\)-conformal connection \({}^{[g]}\nabla \in \mathfrak{A}_{[g]}(M)\) so that \({}^{[g]}\nabla+A_{[g]} \in \mathfrak{A}_{\mathfrak{p}}(M)\). The \(1\)-form \(A_{[g]}\) appeared previously in the work of Matveev & Trautman [35] and may be thought of as the ‘difference’ between \(\mathfrak{p}\) and \([g]\). In particular, if \(M\) is oriented, we obtain a \(\mathrm{Diff}(M)\)-invariant functional \[\mathcal{F}(\mathfrak{p},[g])=\int_{M}|A_{[g]}|^n_g d \mu_g.\] Fixing a projective structure \(\mathfrak{p}\) on \(M\), we may consider the functional \(\mathcal{E}_{\mathfrak{p}}=\mathcal{F}(\mathfrak{p},\cdot)\), which is a functional on the space \(\mathfrak{C}(M)\) of conformal structures on \(M\) only. It is natural to study the infimum of \(\mathcal{E}_{\mathfrak{p}}\) among all conformal structures on \(M\), and to ask whether there is actually a minimising conformal structure which achieves this infimum. This infimum – which may be considered as a measure of how far \(\mathfrak{p}\) deviates from being defined by a conformal connection – is a new global invariant for oriented projective manifolds.

Of particular interest is the case of surfaces where \(\mathcal{E}_{\mathfrak{p}}\) is just the square of the \(L^2\)-norm of \(A_{[g]}\) taken with respect to \([g]\) and this is the case that we study in detail in this article. It turns out that in the surface case the functional \(\mathcal{E}_{\mathfrak{p}}\) also arises from a rather different viewpoint, which simplifies the computation of its variational equations by using the technique of moving frames.

Inspired by the twistorial construction of holomorphic projective structures by Hitchin [19], it was shown in [13], [42] how to construct a ‘twistor space‘ for smooth projective structures. The choice of a projective structure \(\mathfrak{p}\) on an oriented surface \(\Sigma\) induces a complex structure on the total space of the disk bundle \(Z \to \Sigma\) whose sections are conformal structures on \(\Sigma\). In this sense, \(\mathcal{E}_{\mathfrak{p}}([g])\) can be interpreted as measuring the failure of \([g](\Sigma) \subset Z\) to be a holomorphic curve in \(Z\). We proceed to show that \(\mathfrak{p}\) canonically defines an indefinite Kähler-Einstein structure \((h_{\mathfrak{p}},\Omega_{\mathfrak{p}})\) on a certain submanifold \(Y\) of the projectivised holomorphic cotangent bundle \(\mathbb{P}(T^*_{\mathbb{C}}Z^{1,0})\) of \(Z\). Moreover, every conformal structure \([g] : \Sigma \to Z\) admits a lift \(\widetilde{[g]} : \Sigma \to Y\) so that the variational equations can be expressed as follows:

Here we say that \([g]\) is extremal for \(\mathfrak{p}\) if it is a critical point of \(\mathcal{E}_{\mathfrak{p}}\) with respect to compactly supported variations. Moreover, by weakly conformal we mean that there exists a smooth (and possibly vanishing) function \(f\) on \(\Sigma\) so that for some – and hence any – representative metric \(g \in [g]\), we have \(\widetilde{[g]}^*h_{\mathfrak{p}}=fg\). In fact, in the compact case \(\mathcal{E}_{\mathfrak{p}}([g])\) is, up to the topological constant \(-2\pi\chi(\Sigma)\), just the Dirichlet energy of \(\widetilde{[g]}\). As a consequence, we obtain an optimal lower bound:

We then turn to the problem of finding non-trivial examples of projective structures for which \(\mathcal{E}_{\mathfrak{p}}\) admits extremal conformal structures. The conformal connection \({}^{[g]}\nabla\) determined by the choice of a conformal structure \([g]\) on \((\Sigma,\mathfrak{p})\) may equivalently be thought of as a torsion-free connection \(\varphi\) on the principal \(\mathrm{GL}(1,\mathbb{C})\)-bundle of complex linear coframes of \((\Sigma,[g])\). In addition, the \(1\)-form \(A_{[g]}\) turns out to be twice the real part of a section \(\alpha\) of \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\), where \(K_{\Sigma}\) denotes the canonical bundle of \((\Sigma,[g])\). We provide another interpretation of the variational equations by proving that \([g]\) is extremal for \(\mathfrak{p}\) if and only if the quadratic differential \(\nabla_{\varphi}^{\prime\prime}\alpha\) vanishes identically. Here \(\nabla_{\varphi}\) denotes the connection induced by \(\varphi\) on \(K_{\Sigma}^2\otimes \overline{K_{\Sigma}^{*}}\) and \(\nabla^{\prime\prime}_{\varphi}\) its \((0,\! 1)\)-part. Applying the Riemann–Roch theorem, it follows that a projective structure \(\mathfrak{p}\) on the \(2\)-sphere \(S^2\) admits an extremal conformal structure if and only if \(\mathfrak{p}\) is defined by a conformal connection.

While there are no non-trivial critical points for projective structures on the \(2\)-sphere, the situation is quite different for surfaces with negative Euler characteristic. Indeed, the condition of having a vanishing quadratic differential appeared previously in the projective differential geometry literature. In the celebrated paper “Lie groups and Teichmüller space” [21] Hitchin proposed a generalisation of Teichmüller space \(\mathcal{H}_2\) by identifying a connected component \(\mathcal{H}_n\) – nowadays called the Hitchin component – in the space of conjugacy classes of representations of \(\pi_1(\Sigma)\) into \(\mathrm{PSL}(n,\mathbb{R}\)).¹ Here \(\Sigma\) denotes a compact oriented surface whose genus exceeds one. Using the theory of Higgs bundles [20] and harmonic map techniques, Hitchin showed that the choice of a conformal structure \([g]\) on \(\Sigma\) gives an identification \[\mathcal{H}_n\simeq \bigoplus_{\ell=2}^n H^0(\Sigma,K_{\Sigma}^{\ell}).\] Hitchin conjectured that \(\mathcal{H}_3\) is the space of conjugacy classes of monodromy representations of (flat) properly convex projective structures, a fact later confirmed by Choi and Goldman [10] (the geometric interpretation of the Hitchin component for \(n>3\) is a topic of current interest, c.f. [18], [22], [27] for recent results). Teichmüller space being parametrised by holomorphic quadratic differentials, one might ask if there is a unique choice of a conformal structure on \(\Sigma\), so that \(\mathcal{H}_3\) is parametrised in terms of cubic holomorphic differentials only. This is indeed the case, as was shown independently by Labourie [28] and Loftin [34] (see also [2] and [14] for recent work treating the non-compact case and the case of convex polygons, as well as [30] treating the case of a general real split rank \(2\) group). Furthermore, the conformal structure \([g]\) making the quadratic differential vanish is the conformal equivalence class of the so-called Blaschke metric, which arises by realising the universal cover of a properly convex projective surface as a complete hyperbolic affine \(2\)-sphere, see in particular [34].

Calling a conformal structure \([g]\) on \((\Sigma,\mathfrak{p})\) closed, if \(\varphi\) induces a flat connection on \(\Lambda^2(T^*\Sigma)\), we obtain a novel characterisation of properly convex projective structures among flat projective structures:

We conclude with some remarks about the possible relation between our functional and the energy functional on Teichmüller space [12], [29] which one can associate to a representation in the Hitchin component. Finally, as a by-product of our ideas, we obtain a Gauss–Bonnet type identity for oriented projective surfaces, which we briefly discuss in Appendix I.

Acknowledgements

I would like to thank Nigel Hitchin, Gabriel Paternain, Maciej Dunajski, Charles Frances, Karin Melnick and Stefan Rosemann for helpful conversations or correspondence regarding the topic of this article. The author is also grateful to the anonymous referee for helpful remarks. A part of the research for this article was carried out while the author was visiting the Mathematical Institute at the University of Oxford on a postdoctoral fellowship of the Swiss NSF PA00P2_142053. The author would like to thank the Mathematical Institute for its hospitality.