1 Introduction

1.1 Background

A projective manifold is a pair \((M,\mathfrak{p})\) consisting of a smooth manifold \(M\) and a projective structure \(\mathfrak{p}\), that is, an equivalence class of torsion-free connections on the tangent bundle \(TM\), where two such connections are called projectively equivalent if they share the same geodesics up to parametrisation. A projective manifold \((M,\mathfrak{p})\) is called properly convex if it arises as a quotient of a properly convex open set \(\tilde{M}\subset \mathbb{RP}^n\) by a group \(\Gamma\subset \mathrm{PSL}(n+1,\mathbb{R})\) of projective transformations which acts discretely and properly discontinuously. The geodesics of \(\mathfrak{p}\) are the projections to \(M=\Gamma\!\setminus\!\tilde{M}\) of the projective line segments contained in \(\tilde{M}\). In particular, locally the geodesics of a properly convex projective structure \(\mathfrak{p}\) can be mapped diffeomorphically to segments of straight lines, that is, \(\mathfrak{p}\) is flat.

It follows from the work of Cheng–Yau [9, 10] that the universal cover \(\tilde{M}\) of a properly convex projective manifold \((M,\mathfrak{p})\) determines a unique properly embedded hyperbolic affine sphere \(f : \tilde{M} \to \mathbb{R}^{n+1}\), which is asymptotic to the cone over \(\tilde{M}\) in \(\mathbb{R}^{n+1}\). The Blaschke metric and Blaschke connection induced by \(f\) descend to the quotient \(\Gamma\!\setminus\!\tilde{M}\) and equip \(M\) with a complete Riemannian metric \(g\) and projectively flat connection \(\nabla \in \mathfrak{p}\), see the work of Loftin [30]. The difference between \(\nabla\) and the Levi-Civita connection of the Blaschke metric is encoded in terms of a cubic form, the so-called Fubini–Pick form of \(f\). For an introduction to affine differential geometry the reader may consult [33] as well as [29] for a nice survey on affine spheres.

Properly convex projective surfaces are of particular interest, as they may be seen – through the work of Hitchin [25], Goldman [20] and Choi–Goldman [12] – as the natural generalisation of the notion of a hyperbolic Riemann surface. In the case of a properly convex oriented surface \((\Sigma,\mathfrak{p})\), the Fubini–Pick form is the real part of a cubic differential that is holomorphic with respect to the Riemann surface structure on \(\Sigma\) defined by the orientation and the conformal equivalence class of the Blaschke metric. Conversely, Wang [36] observed that a holomorphic cubic differential \(C\) on a closed hyperbolic Riemann surface \((\Sigma,[g])\) determines a unique conformal Riemannian metric \(g\) whose Gauss curvature \(K_g\) satisfies \[\tag{1.1} K_g=-1+2|C|^2_g,\] where \(|C|_g\) denotes the point-wise tensor norm of \(C\) with respect to the Hermitian metric induced by \(g\) on the third power of the canonical bundle of \(\Sigma\). Furthermore, the pair \((g,\operatorname{Re}(C))\) can be realized as the Blaschke metric and Fubini–Pick form of a complete hyperbolic affine sphere \(f : \tilde{M} \to \mathbb{R}^3\) defined on the universal cover \(\tilde{M}\) of \(M\). In particular, combining Wang’s work with the work of Loftin establishes – on a compact oriented surface of negative Euler characteristic – a bijective correspondence between properly convex projective structures and pairs \(([g],C)\) consisting of a conformal structure \([g]\) and a cubic holomorphic differential \(C\), see [30]. This correspondence was also discovered independently by Labourie [26]. Since then, Benoist–Hulin [1] have extended the correspondence to noncompact projective surfaces with finite Finsler volume and Dumas–Wolf [14] study the case of polynomial cubic differentials on the complex plane.

In [28], Libermann constructs a para-Kähler structure \((h_0,\Omega_0)\) on the open submanifold \(A_0\subset \mathbb{RP}^n\times \mathbb{RP}^{n*}\) consisting of non-incident point-line pairs. A para-Kähler structure may be thought of as a split-complex analogue of the notion of a Kähler structure. In particular, \(h_0\) is a pseudo-Riemannian metric of split-signature \((n,n)\) and \(\Omega_0\) a symplectic form, so that there exists an endomorphism of the tangent bundle relating \(h_0\) and \(\Omega_0\) which squares to become the identity map. In [22, 23], Hildebrand – see also the related work [17, 19, 35] – observed that proper affine spheres \(f : M \to \mathbb{R}^{n+1}\) correspond to minimal Lagrangian immersions \(\hat{f} : M \to A_0\). Thus, the result of Hildebrand, combined with the work of Cheng–Yau, associates a minimal Lagrangian immersion to every properly convex projective manifold.

1.2 Minimal Lagrangian connections

Here we propose a generalization of the notion of a properly convex projective surface which arises naturally from the concept of a minimal Lagrangian connection. In joint work with Dunajski the author has shown that the construction of Libermann is a special case of a more general result: In [15], it is shown that a projective structure \(\mathfrak{p}\) on an \(n\)-manifold \(M\) canonically defines an almost para–Kähler structure \((h_{\mathfrak{p}},\Omega_{\mathfrak{p}})\) on the total space of a certain affine bundle \(A\to M\), whose underlying vector bundle is the cotangent bundle of \(M\). The bundle \(A \to M\) has the crucial property that its sections are in one-to-one correspondence with the representative connections of \(\mathfrak{p}\). Therefore, fixing a representative connection \(\nabla \in \mathfrak{p}\) gives a section \(s_{\nabla} : M \to A\) and hence an isomorphism \(\psi_{\nabla} : T^*M \to A\), by declaring the origin of the affine fibre \(A_p\) to be \(s_{\nabla}(p)\) for all \(p \in M\). Correspondingly, we obtain a pair \((h_{\nabla},\Omega_{\nabla})=\psi_{\nabla}^*(h_{\mathfrak{p}},\Omega_{\mathfrak{p}})\) on the total space of the cotangent bundle. Besides being a geometric structure of interest in itself (see [15] for details), the pair \((h_{\nabla},\Omega_{\nabla})\) has the natural property \[o^*h_{\nabla}=(s_{\nabla})^*h_{\mathfrak{p}}=-\left(\frac{1}{n-1}\right)\mathrm{Ric}^+(\nabla)\] and \[o^*\Omega_{\nabla}=(s_{\nabla})^*\Omega_{\mathfrak{p}}=\left(\frac{1}{n+1}\right)\mathrm{Ric}^{-}(\nabla),\] where \(o : M \to T^*M\) denotes the zero-section and \(\mathrm{Ric}^{\pm}(\nabla)\) the symmetric (respectively, the anti-symmetric) part of the Ricci curvature \(\mathrm{Ric}(\nabla)\) of \(\nabla\). Consequently, we call \(\nabla\) Lagrangian if the Ricci tensor of \(\nabla\) is symmetric, or equivalently, if the zero-section \(o\) is a Lagrangian submanifold of \((T^*M,\Omega_{\nabla}\)). Likewise, we call \(\nabla\) timelike/spacelike if \(\pm\mathrm{Ric}^+(\nabla)\) is positive definite, or equivalently, if the zero-section \(o\) is a timelike/spacelike submanifold of \((T^*M,h_{\nabla})\). The upper sign corresponds to the timelike case and lower sign to the spacelike case. Moreover, we call \(\nabla\) minimal if the zero-section is a minimal submanifold of \((T^*M,h_{\nabla})\).

We henceforth restrict our considerations to the case of oriented surfaces. We show (see Theorem 4.4 below) that a timelike/spacelike Lagrangian connection \(\nabla\) is minimal if and only if \[R^{ij}\left(2\nabla_iR_{jk}-\nabla_k R_{ij}\right)=0,\] where \(R_{ij}\) denotes the Ricci tensor of \(\nabla\) and \(R^{ij}\) its inverse. We then show that a minimal Lagrangian connection \(\nabla\) on an oriented surface \(\Sigma\) defines a triple \((g,\beta,C)\) on \(\Sigma\), consisting of a Riemannian metric \(g\), a \(1\)-form \(\beta\) and a cubic differential \(C\), so that the following equations hold

\[\tag{1.2} K_g=\pm 1+2\,|C|_g^2+\delta_g\beta, \qquad \overline{\partial} C=\left(\beta-\mathrm{i}\star_g \beta\right)\otimes C,\qquad \mathrm{d}\beta=0.\] As usual, \(\mathrm{i}=\sqrt{-1}\), \(\overline{\partial}\) denotes the “del-bar” operator with respect to the integrable almost complex structure \(J\) induced on \(\Sigma\) by \([g]\) and the orientation, \(\star_g\), \(\delta_g\) and \(K_g\) denote the Hodge-star, co-differential and Gauss curvature with respect to \(g\), respectively. Recall that \(|C|_g\) denotes the point-wise tensor norm of \(C\) with respect to the Hermitian metric induced by \(g\) on the third power of the canonical bundle of \(\Sigma\). As a consequence, we use a result of Labourie [26] to prove that if \(\nabla\) is a spacelike minimal Lagrangian connection on a compact oriented surface \(\Sigma\) defining a flat projective structure \(\mathfrak{p}(\nabla)\), then \((\Sigma,\mathfrak{p})\) is a properly convex projective surface. Moreover, the zero-section is a totally geodesic submanifold of \((T^*\Sigma,h_{\nabla})\) if and only if \(\nabla\) is the Levi-Civita connection of a hyperbolic metric.

We also show that a minimal Lagrangian connection defines a flat projective structure if and only if \(\beta\) vanishes identically. In particular, we recover Wang’s equation (1.1) in the projectively flat case. In the projectively flat case Labourie [26] interpreted the first two equations as an instance of Hitchin’s Higgs bundle equations [24]. In the case with \(\beta\neq 0\) the above triple of equations falls into the general framework of symplectic vortex equations [13] (see also [3]). Furthermore, it appears likely that the above equations also admit an interpretation in terms of affine differential geometry, but this will be addressed elsewhere.

The last two of the equations (1.2) say that locally there exists a (real-valued) function \(r\) so that \(\mathrm{e}^{-2r}C\) is holomorphic. As a consequence of this we show that the only examples of minimal Lagrangian connections on the \(2\)-sphere are Levi-Civita connections of metrics of positive Gauss curvature.

Furthermore, if \((\Sigma,[g])\) is a compact Riemann surface of negative Euler characteristic \(\chi(\Sigma)\), then the metric \(g\) of the triple \((g,\beta,C)\) is uniquely determined in terms of \(([g],\beta,C)\). This leads to a quasi-linear elliptic PDE of vortex type, which belongs to a class of equations solved in [17] using the technique of sub – and supersolutions (see also [14] for the case when \(\beta\) vanishes identically). Here instead, we use the calculus of variations and prove existence and uniqueness of a smooth minimum of the following functional defined on the Sobolev space \(W^{1,2}(\Sigma)\) \[\mathcal{E}_{\kappa,\tau} : W^{1,2}(\Sigma) \to \overline{\mathbb{R}}, \quad u\mapsto \frac{1}{2}\int_{\Sigma}|\mathrm{d}u|^2_{g_0}-2u-\kappa\mathrm{e}^{2u}+\tau\mathrm{e}^{-4u}dA_{g_0},\] where \(\kappa,\tau \in C^{\infty}(\Sigma)\) satisfy \(\kappa<0\), \(\tau \geqslant 0\) and \(g_0\) denotes the hyperbolic metric in the conformal equivalence class \([g]\).

An immediate consequence of (1.2) is that the area of a spacelike minimal Lagrangian connection \(\nabla\) – by which we mean the area of \(o(\Sigma)\subset (T^*\Sigma,h_{\nabla})\) – satisfies the inequality \[\mathrm{Area}(\nabla)\geqslant-2\pi\chi(\Sigma).\] Therefore, we call a spacelike minimal Lagrangian connection with area \(-2\pi\chi(\Sigma)\) area minimising. We obtain:

Using the classification of properly convex projective structures by Loftin [30] and Labourie [26], it follows that every properly convex projective structure on \(\Sigma\) arises from a unique spacelike minimal Lagrangian connection, paralleling the result of Hildebrand.

1.3 Related work

After the first version of this article appeared on the arXiv, Daniel Fox informed the author about his interesting paper [17], which contains some closely related results. Here we briefly compare our results which were arrived at independently. In a previous preprint [16] (see also [18]), Fox introduced the notion of an AH (affine hypersurface) structure which is a pair comprising a projective structure and a conformal structure satisfying a compatibility condition which is automatic in two dimensions. He then proceeds to postulate Einstein equations for AH structures, which are motivated by Calderbank’s work on Einstein–Weyl structures on surfaces [5]. Subsequently in [17], Fox classifies the Einstein AH structures on closed oriented surfaces and in particular, observes that in the case of negative Euler-characteristic they precisely correspond to the properly convex projective structures. On the \(2\)-sphere \(S^2\) he recovers the Einstein–Weyl structures of Calderbank. On \(S^2\), the Einstein AH structures and the minimal Lagrangian connections “overlap” in the space of metrics of constant positive Gauss–curvature. In the case of negative Euler characteristic, the minimal Lagrangian connections are however strictly more general than the Einstein AH structures. Indeed, in this case, the projective structures arising from minimal Lagrangian connections provide a new and previously unstudied class of projective structures.

This new class of (possibly) curved projective structures may be thought of as a generalization of the notion of a (flat) properly convex projective structure. One would expect that this class exhibits interesting properties, similar to those of properly convex projective structures. As a first result in this direction, it is shown in [32], that the geodesics of a minimal Lagrangian connection naturally give rise to a flow admitting a dominated splitting (a certain weakening of the notion of an Anosov flow). In particular, this flow provides a generalization of the geodesic flow induced by the Hilbert metric on the quotient surface of a divisible convex set.

Furthermore, in joint work in progress by the author and A. Cap [8], the notion of a minimal Lagrangian connection is extended to all so-called \(|1|\)-graded parabolic geometries. This is a class of geometric structures which, besides projective geometry, includes (but is not restricted to) conformal geometry, (almost) Grassmannian geometry and (almost) quaternionic geometry.

Acknowledgements

The author is grateful to Andreas Cap, Daniel Fox, Gabriel Paternain, Nigel Hitchin, Norbert Hungerbühler, Tobias Weth and Luca Galimberti for helpful conversations or correspondence. The author also would like to thank the anonymous referees for several helpful suggestions.