Minimal Lagrangian connections on compact surfaces

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 [35] 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 [38] 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 \[ K_g=-1+2|C|^2_g,\](1.1) 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, 37] – 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 4.3 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

\[ K_g=\pm 1+2\,|C|_g^2+\delta_g\beta, \qquad \overline{\partial} C=\left(\beta-\mathrm{i}\star_g \beta\right)\otimes C,\qquad \mathrm{d}\beta=0.\](1.2) 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 eprint [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 [34], 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.

2 Preliminaries

Throughout the article \(\Sigma\) will denote an oriented smooth \(2\)-manifold without boundary. All manifolds and maps are assumed to be smooth and we adhere to the convention of summing over repeated indices.

2.1 The coframe bundle

We denote by \(\upsilon : F \to \Sigma\) the bundle of orientation preserving coframes whose fibre at \(p \in \Sigma\) consists of the linear isomorphisms \(f : T_p\Sigma \to \mathbb{R}^2\) that are orientation preserving with respect to the fixed orientation on \(\Sigma\) and the standard orientation on \(\mathbb{R}^2\). Recall that \(\upsilon : F \to \Sigma\) is a principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle with right action defined by the rule \(R_a(f)=f\cdot a=a^{-1}\circ f\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\). The bundle \(F\) is equipped with a tautological \(\mathbb{R}^2\)-valued \(1\)-form \(\omega=(\omega^i)\) defined by \(\omega_f=f\circ \upsilon^{\prime}_f\), and this \(1\)-form satisfies the equivariance property \(R_a^*\omega=a^{-1}\omega\). A torsion-free connection \(\nabla\) on \(T\Sigma\) corresponds to a \(\mathfrak{gl}(2,\mathbb{R}\))-valued connection \(1\)-form \(\theta=(\theta^i_j)\) on \(F\) satisfying the structure equations \[\begin{aligned} \mathrm{d}\omega&=-\theta\wedge \omega,\\ \mathrm{d}\theta&=-\theta\wedge\theta+\Theta, \end{aligned}\](2.1)–(2.2) where \(\Theta\) denotes the curvature \(2\)-form of \(\theta\). The Ricci curvature of \(\nabla\) is the (not necessarily symmetric) covariant \(2\)-tensor field \(\mathrm{Ric}(\nabla)\) on \(\Sigma\) satisfying \[\mathrm{Ric}(\nabla)(X,Y)=\operatorname{tr}\left( Z\mapsto \nabla_Z\nabla_X Y-\nabla_X\nabla_Z Y-\nabla_{[Z,X]}Y\right), \quad Z\in \Gamma(TM),\] for all vector fields \(X,Y\) on \(\Sigma\). Denoting by \(\mathrm{Ric}^{\pm}(\nabla)\) the symmetric (respectively, the anti-symmetric) part of the Ricci curvature of \(\nabla\), so that \(\mathrm{Ric}(\nabla)=\mathrm{Ric}^{+}(\nabla)+\mathrm{Ric}^{-}(\nabla)\), the (projective) Schouten tensor of \(\nabla\) is defined as \[\mathrm{Schout}(\nabla)=\mathrm{Ric}^+(\nabla)-\frac{1}{3}\mathrm{Ric}^{-}(\nabla).\] Since the components of \(\omega\) are a basis for the \(\upsilon\)-semibasic forms on \(F\),¹ it follows that there exist real-valued functions \(S_{ij}\) on \(F\) such that \[\upsilon^*\mathrm{Schout}(\nabla)=\omega^tS\omega=S_{ij}\omega^i\otimes \omega^j,\] where \(S=(S_{ij})\). Note that \[R_a^*S=a^tSa\] for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\), since \(\omega^tS\omega\) is invariant under \(R_a\). In terms of the functions \(S_{ij}\) the curvature \(2\)-form \(\Theta=(\Theta^i_j)\) can be written as² \[ \Theta^i_j=\left(\delta^i_{[k}S_{l]j}-\delta^i_jS_{[kl]}\right)\omega^k\wedge\omega^l,\](2.3) or explicitly \[ \Theta=\begin{pmatrix} 2S_{21}-S_{12} & S_{22}\\ -S_{11} & S_{21}-2S_{12}\end{pmatrix}\omega^1\wedge\omega^2.\](2.4)

2.2 The orthonormal coframe bundle

Recall that if \(g\) is a Riemannian metric on the oriented surface \(\Sigma\), the Levi-Civita connection \((\varphi^i_j)\) of \(g\) is the unique connection on the coframe bundle \(\upsilon : F \to \Sigma\) satisfying \[\begin{aligned} \mathrm{d}\omega^i&=-\varphi^i_j\wedge\omega^j,\\ \mathrm{d}g_{ij}&=g_{ik}\varphi^k_j+g_{kj}\varphi^k_i, \end{aligned}\] where we write \(\upsilon^*g=g_{ij}\omega^i\otimes\omega^j\) for real-valued functions \(g_{ij}=g_{ji}\) on \(F\). Differentiating these equations implies that there exists a unique function \(K_g\), the Gauss curvature of \(g\), so that \[\mathrm{d}\varphi^i_j+\varphi^i_k\wedge\varphi^k_j=g_{jk}K_g\omega^i\wedge\omega^k.\] We may reduce \(F\) to the \(\mathrm{SO}(2)\)-subbundle \(F_g\) consisting of orientation preserving coframes that are also orthonormal with respect to \(g\) , that is, the bundle defined by the equations \(g_{ij}=\delta_{ij}\). On \(F_g\) the identity \(\mathrm{d}g_{ij}= 0\) implies the identities \(\varphi^1_1=\varphi^2_2= 0\) as well as \(\varphi^1_2+\varphi^2_1= 0\). Therefore, writing \(\varphi:=\varphi^2_1\), we obtain the structure equations \[\begin{aligned} \begin{split} \mathrm{d}\omega_1&=-\omega_2\wedge\varphi,\\ \mathrm{d}\omega_2&=-\varphi\wedge\omega_1,\\ \mathrm{d}\varphi&=-K_g\omega_1\wedge\omega_2, \end{split} \end{aligned}\](2.5) where \(\omega_i=\delta_{ij}\omega^j\). Continuing to denote the basepoint projection \(F_g \to \Sigma\) by \(\upsilon\), the area form \(dA_g\) of \(g\) satisfies \(\upsilon^*dA_g=\omega_1\wedge\omega_2\). Also, note that a complex-valued \(1\)-form \(\alpha\) on \(\Sigma\) is a \((1,\! 0)\)-form for the complex structure \(J\) induced on \(\Sigma\) by \(g\) and the orientation if and only if \(\upsilon^*\alpha\) is a complex multiple of the complex-valued form \(\omega=\omega_1+\mathrm{i}\omega_2\). In particular, denoting by \(K_{\Sigma}\) the canonical bundle of \(\Sigma\) with respect to \(J\), a section \(A\) of the \(\ell\)-th tensorial power of \(K_{\Sigma}\) satisfies \(\upsilon^*A=a\omega^{\ell}\) for some unique complex-valued function \(a\) on \(F_g\). Denote by \(S^3_0(T^*\Sigma)\) the trace-free part of \(S^3(T^*\Sigma)\) with respect to \([g]\), where \(S^3(T^*\Sigma)\) denotes the third symmetrical power of the cotangent bundle of \(\Sigma\). The proof of the following lemma is an elementary computation and thus omitted.

In complex notation, the structure equations of a cubic differential \(C \in \Gamma(K_{\Sigma}^3)\) can be written as follows. Writing \(\upsilon^*C=c\omega^3\) for a complex-valued function \(c\) on \(F_g\), it follows from the \(\mathrm{SO}(2)\)-equivariance of \(c\omega^3\) that there exist complex-valued functions \(c^{\prime}\) and \(c^{\prime\prime}\) on \(F_g\) such that \[\mathrm{d}c=c^{\prime}\omega+c^{\prime\prime}\overline{\omega}+3\mathrm{i}c\varphi,\] where we write \(\overline{\omega}=\omega_1-\mathrm{i}\omega_2\). Note that the Hermitian metric induced by \(g\) on \(K_{\Sigma}^3\) has Chern connection \(\mathrm{D}\) given by \[c \mapsto \mathrm{d}c-3\mathrm{i}c\varphi.\] In particular, the \((0,\! 1)\)-derivative of \(C\) with respect to \(\mathrm{D}\) is represented by \(c^{\prime\prime}\), that is, \(\upsilon^*(\mathrm{D}^{0,1}C)=c^{\prime\prime}\omega^3\otimes \overline{\omega}\). Since \(\overline{\partial}=\mathrm{D}^{0,1}\), we obtain \[ \upsilon^*\left(\overline{\partial}C\right)=c^{\prime\prime}\omega^3\otimes\overline{\omega}.\](2.6) Also, we record the identity \[\upsilon^*|C|_g^2=|c|^2.\]

Moreover, recall that for \(u \in C^{\infty}(\Sigma)\) we have the following standard identity for the change of the Gauss curvature of a metric \(g\) under conformal rescaling \[K_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\left(K_g-\Delta_g u\right),\] where \(\Delta_g=-\left(\delta_g \mathrm{d}+\mathrm{d}\delta_g\right)\) is the negative of the Laplace–Beltrami operator with respect to \(g\). Also, \[dA_{\mathrm{e}^{2u}g}=\mathrm{e}^{2u}dA_g\] for the change of the area form \(dA_g\), \[\Delta_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\Delta_g\] for \(\Delta_g\) acting on functions and \[\delta_{\mathrm{e}^{2u}g}=\mathrm{e}^{-2u}\delta_g\] for the co-differential acting on \(1\)-forms. Finally, the norm of \(C\) changes as \[|C|^2_{\mathrm{e}^{2u}g}=\mathrm{e}^{-6u}|C|^2_{g}.\]

2.3 The cotangent bundle and induced structures

Recall that we have a \(\mathrm{GL}^+(2,\mathbb{R})\) representation \(\varrho\) on \(\mathbb{R}_2\) – the real vector space of row vectors of length two with real entries – defined by the rule \(\varrho(a) \xi=\xi a^{-1}\) for all \(\xi \in \mathbb{R}_2\) and \(a \in \mathrm{GL}^+(2,\mathbb{R})\). The cotangent bundle of \(\Sigma\) is the vector bundle associated to the coframe bundle \(F\) via the representation \(\varrho\), that is, the bundle obtained by taking the quotient of \(F\times \mathbb{R}_2\) by the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action induced by \(\varrho\). Consequently, a \(1\)-form on \(\Sigma\) is represented by an \(\mathbb{R}_2\)-valued function \(\xi\) on \(F\) which is \(\mathrm{GL}^+(2,\mathbb{R})\)-equivariant, that is, \(\xi\) satisfies \(R_a^*\xi=\xi a\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\).

Using \(\theta\) we may define a Riemannian metric \(h_{\nabla}\) as well as a symplectic form \(\Omega_{\nabla}\) on \(T^*\Sigma\) as follows. Let \[\pi : F\times \mathbb{R}_2 \to \left(F\times \mathbb{R}_2\right)/\mathrm{GL}^+(2,\mathbb{R})\simeq T^*\Sigma\] denote the quotient projection. Writing \[\psi=\mathrm{d}\xi-\xi\theta-\xi\omega\xi-\omega^tS^t\] or in components \(\psi=(\psi_i)\) with \[\psi_i=\mathrm{d}\xi_i-\xi_j\theta^j_i-\xi_j\omega^j\xi_i-S_{ji}\omega^j,\] we consider the covariant \(2\)-tensor field \(T_{\nabla}=\psi\omega:=\psi_i\otimes \omega^i\). Note that the \(\pi\)-semibasic \(1\)-forms on \(F\times \mathbb{R}_2\) are given by the components of \(\omega\) and \(\mathrm{d}\xi-\xi\theta\), or, equivalently, by the components of \(\omega\) and \(\psi\). Indeed, the components of \(\omega\) are semibasic by the definition of \(\omega\). Moreover, if \(X_v\) for \(v \in \mathfrak{gl}{(2,\mathbb{R})}\) is a fundamental vector field for the coframe bundle \(F \to \Sigma\), that is, the vector field associated to the flow \(R_{\exp(tv)}\), then \[\begin{aligned} (\mathrm{d}\xi-\xi\theta)(X_v)&=-\xi\theta(X_v)+\lim_{t\to 0}\frac{1}{t}\left((R_{\exp(tv)})^*\xi-\xi\right)\\ &=-\xi v+\lim_{t\to 0}\frac{1}{t}\left(\xi\exp(tv)-\xi\right)=-\xi v+\xi v=0, \end{aligned}\] where we have used that the connection \(\theta\) maps a fundamental vector field \(X_v\) to its generator \(v \in \mathfrak{gl}(2,\mathbb{R})\). Since the fundamental vector fields span the vector fields tangent to fibres of \(\pi\), it follows that the components of \(\mathrm{d}\xi-\xi\theta\) are \(\pi\)-semibasic. Moreover, \[\begin{aligned} R_a^*\psi&=\mathrm{d}\xi\, a-\xi a a^{-1}\theta a-\xi a a^{-1}\omega\xi a-\omega^t (a^{-1})^t a^tS^ta=\psi a,\\ R_a^*\omega&=a^{-1}\omega, \end{aligned}\](2.7)–(2.8) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\), it follows that the \(\pi\)-semibasic tensor field \(T_{\nabla}\) is invariant under the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action and hence there exists a unique symmetric covariant \(2\)-tensor field \(h_{\nabla}\) and a unique anti-symmetric covariant \(2\)-tensor field \(\Omega_{\nabla}\) on \(T^*\Sigma\) such that \[\pi^*\left(h_{\nabla}+\Omega_{\nabla}\right)=T_{\nabla}.\] Using the structure equation (2.1), we compute \[\begin{aligned} \pi^*\Omega_{\nabla}&=\mathrm{d}\xi_i\wedge \omega^i-\xi_j\theta^j_i\wedge\omega^i-\xi_i\xi_j\omega^j\wedge\omega^i-S_{ji}\omega^j\wedge\omega^i\\ &=\mathrm{d}\xi_i\wedge\omega^i+\xi_i\mathrm{d}\omega^i-S_{ij}\omega^i\wedge\omega^j\\ &=\mathrm{d}(\xi_i\omega^i)-S_{[ij]}\omega^i\wedge\omega^j. \end{aligned}\] The \(1\)-form \(\xi\omega=\xi_i\omega^i\) on \(F\times \mathbb{R}_2\) is \(\pi\)-semibasic and \(R_a\) invariant, hence the \(\pi\)-pullback of a unique \(1\)-form \(\tau\) on \(T^*\Sigma\) which is the tautological \(1\)-form of \(T^*\Sigma\). Recall that the canonical symplectic form on \(T^*\Sigma\) is \(\Omega_0=\mathrm{d}\tau\), hence \(\Omega_{\nabla}\) defines a symplectic structure on \(T^*\Sigma\) which is the canonical symplectic structure twisted with the (closed) \(2\)-form \(\frac{1}{3}\mathrm{Ric}^{-}(\nabla)\) \[\Omega_{\nabla}=\Omega_0+\Upsilon^*\left(\frac{1}{3}\mathrm{Ric}^{-}(\nabla)\right),\] where \(\Upsilon : T^*\Sigma \to \Sigma\) denotes the basepoint projection. In particular, denoting by \(o : \Sigma \to T^*\Sigma\) the zero \(\Upsilon\)-section, the definition of the Schouten tensor gives \[o^*\Omega_{\nabla}=\frac{1}{3}\mathrm{Ric}^{-}(\nabla).\] This shows:

Which motivates:

Also, we obtain for the symmetric part \[\pi^*h_{\nabla}=\psi\circ \omega:=\psi_1\circ \omega^1+\psi_2\circ \omega^2\] where \(\circ\) denotes the symmetric tensor product. Since the four \(1\)-forms \(\psi_1,\psi_2,\omega^1,\omega^2\) are linearly independent, it follows that \(h_{\nabla}\) is non-degenerate and hence defines a pseudo-Riemannian metric of split signature \((1,1,-1,-1)\) on \(T^*\Sigma\).

From the definition of the Schouten tensor and \(h_{\nabla}\) we immediately obtain \[ o^*h_{\nabla}=-\mathrm{Ric}^+(\nabla).\](2.9) Following standard pseudo-Riemannian submanifold theory, we call a tangent vector \(v\) timelike if \(h_{\nabla}(v,v)<0\) and spacelike if \(h_{\nabla}(v,v)>0\). Thus (2.9) motivates:

3 Twisted Weyl connections

We will see that timelike/spacelike minimal Lagrangian connections are twisted Weyl connections. In this section we study some properties of this class of connections that we will need later during the classification of spacelike minimal Lagrangian connections.

Let \([g]\) be a conformal structure on the smooth oriented surface \(\Sigma\). By a \([g]\)-Weyl connection on \(\Sigma\) we mean a torsion-free connection on \(T\Sigma\) preserving the conformal structure \([g]\). It follows from Koszul’s identity that a \([g]\)-Weyl connection can be written in the following form \[{}^{(g,\beta)}\nabla={}^g\nabla+g\otimes \beta^{\sharp}-\beta\otimes\mathrm{Id}-\mathrm{Id}\otimes \beta,\] where \(g \in [g]\), \(\beta \in \Omega^1(\Sigma)\) is a \(1\)-form and \(\beta^{\sharp}\) denotes the \(g\)-dual vector field to \(\beta\). We will use the notation \({}^{[g]}\nabla\) to denote a general \([g]\)-Weyl connection.

Note that if \(\alpha\) satisfies the above properties, then \({}^{[g]}\nabla+\alpha\) is torsion-free. Moreover, the covariant \(3\)-tensor obtained by lowering the upper index of \(\alpha\) with a metric \(g \in [g]\) gives a section of \(\Gamma(S^3_0(T^*\Sigma))\). Conversely, every \(\mathrm{End}(T\Sigma)\)-valued \(1\)-form on \(\Sigma\) satisfying the above properties arises in this way. In other words, fixing a Riemannian metric \(g \in [g]\) allows to identify the twist term \(\alpha\) with a cubic differential.

Fixing a metric \(g\in [g]\), the connection form \(\theta=(\theta^i_j)\) of a twisted Weyl connection is given by \[\theta^i_j=\varphi^i_j+\left(b_kg^{ki}g_{jl}-\delta^i_jb_l-\delta^i_lb_j+a^i_{jl}\right)\omega^l,\] where the map \((g_{ij}) : F \to S^2(\mathbb{R}_2)\) represents the metric \(g\), the map \((b_i) : F \to \mathbb{R}_2\) represents the \(1\)-form \(\beta\) and the map \((a^i_{jk}) : F \to \mathbb{R}^2\otimes S^2(\mathbb{R}_2)\) represents the \(1\)-form \(\alpha\). Moreover, \((\varphi^i_j)\) denote the Levi-Civita connection forms of \(g\). Reducing to the bundle \(F_g\) of \(g\)-orthonormal orientation preserving coframes, the connection form becomes \[\theta=\begin{pmatrix} -\beta & \star_g\beta-\varphi \\ \varphi-\star_g\beta & -\beta\end{pmatrix}+\begin{pmatrix} a^1_{11}\omega_1+a^1_{12}\omega_2 & a^1_{12}\omega^1+a^1_{22}\omega_2 \\ a^2_{11}\omega_1+a^2_{12}\omega_2 & a^2_{12}\omega_1+a^2_{22}\omega_2\end{pmatrix},\] where we use the identity \(\upsilon^*\left(\star_g\beta\right)=-b_2\omega_1+b_1\omega_2\). By definition, on \(F_g\) the functions \(a^i_{jk}\) satisfy the identities \[a^{i}_{jk}=a^i_{kj}, \qquad a^{k}_{kj}=0, \qquad \delta_{ki}a^{k}_{jl}=\delta_{kj}a^k_{il}.\] Thus, writing \(c_1=a^1_{11}\) and \(c_2=a^2_{22}\), we obtain \[\theta=\begin{pmatrix} -\beta & \star_g \beta-\varphi \\ \varphi-\star_g\beta & -\beta\end{pmatrix}+\begin{pmatrix} c_1\omega_1-c_2\omega_2 & -c_2\omega_1-c_1\omega_2\\ -c_2\omega_1-c_1\omega_2 & -c_1\omega_1+c_2 \omega_2\end{pmatrix}.\] In order to compute the curvature form of \(\theta\) we first recall that we write \(\upsilon^*\beta=b_i\omega_i\) and since \(b_i\omega_i\) is \(\mathrm{SO}(2)\)-invariant, it follows that there exist unique real-valued functions \(b_{ij}\) on \(F_g\) such that \[\begin{aligned} \mathrm{d}b_1&=b_{11}\omega_1+b_{12}\omega_2+b_2\varphi,\\ \mathrm{d}b_2&=b_{21}\omega_1+b_{22}\omega_2-b_1\varphi. \end{aligned}\] Recall also that the area form of \(g\) satisfies \(\upsilon^*dA_g=\omega_1\wedge\omega_2\) and since \(\star_g 1=dA_g\), we get \[\upsilon^*\delta_g \beta=-(b_{11}+b_{22}),\] as well as \[\upsilon^*\left(\mathrm{d}\!\star_g\!\beta\right)=(b_{11}+b_{22})\omega_1\wedge\omega_2.\] Since \(c_1+\mathrm{i}c_2\) represents a cubic differential on \(\Sigma\), there exist unique real-valued functions \(c_{ij}\) on \(F_g\) such that \[\begin{aligned} \mathrm{d}c_1&=c_{11}\omega_1+c_{12}\omega_2-3c_2\varphi,\\ \mathrm{d}c_2&=c_{21}\omega_1+c_{22}\omega_2+3c_1\varphi. \end{aligned}\] Consequently, a straightforward calculation shows that the curvature form \(\Theta=\mathrm{d}\theta+\theta\wedge\theta\) satisfies \[\begin{gathered} \Theta=\begin{pmatrix} -\mathrm{d}\beta & K_gdA_g+\mathrm{d}\star_g\beta-\frac{1}{2}|\alpha|^2_g\omega_1\wedge\omega_2\\ -K_gd A_g- \mathrm{d}\star_g\beta+\frac{1}{2}|\alpha|^2_g\omega_1\wedge\omega_2& -\mathrm{d}\beta \end{pmatrix}\\ + \begin{pmatrix} 2(b_1c_2+b_2c_1)-(c_{12}+c_{21}) & 2(b_1c_1-b_2c_2)+\left(c_{22}-c_{11}\right)\\ 2(b_1c_1-b_2c_2)+\left(c_{22}-c_{11}\right) & 2(-b_1c_2-b_2c_1)+(c_{12}+c_{21})\end{pmatrix} \omega_1\wedge\omega_2, \end{gathered}\](3.1) where we use the identity \(\upsilon^*|\alpha|^2_g=4\left((c_1)^2+(c_2)^2\right)\).

3.1 A characterisation of twisted Weyl connections

We obtain a natural differential operator \(\mathrm{D}_{[g]}\) acting on the space \(\mathfrak{A}(\Sigma)\) of torsion-free connections on \(T\Sigma\) \[\mathrm{D}_{[g]} : \mathfrak{A}(\Sigma) \to \Omega^2(\Sigma), \quad \nabla \mapsto \operatorname{tr}_g \mathrm{Ric}(\nabla)dA_g.\] Note that this operator does indeed only depend on the conformal equivalence class of \(g\). A twisted Weyl connection \(\nabla\) on \((\Sigma,[g])\) can be characterised by minimising the integral of \(\mathrm{D}_{[g]}\) among its projective equivalence class \(\mathfrak{p}(\nabla)\).

In [32] the following result is shown, albeit phrased in different language:

Let \(\mathfrak{P}(\Sigma)\) denote the space of projective structures on \(\Sigma\). Using Proposition 3.2 and Proposition 3.3 we immediately obtain:

4 Submanifold theory of Lagrangian connections

We now restrict attention to torsion-free connections on \(T\Sigma\) having symmetric Ricci tensor, so that the zero-section \(o : \Sigma \to T^*\Sigma\) is a Lagrangian submanifold. If furthermore \(\nabla\) is timelike/spacelike, we obtain an induced metric \(g=\mp o^*h_{\nabla}=\pm \mathrm{Ric}(\nabla)\) and the immersion \(o : \Sigma \to T^*\Sigma\) has a well-defined normal bundle and second fundamental form. In particular, we want to compute when \(o : \Sigma \to T^*\Sigma\) is minimal, that is, the trace with respect to \(g\) of its second fundamental form vanishes identically.

4.1 Algebraic preliminaries

Before we delve into the computations, we briefly review the relevant algebraic structure of the theory of oriented surfaces in an oriented Riemannian – and oriented split-signature Riemannian four manifold \((M,g)\). We refer the reader to [4] and [11] for additional details.

First, let \(X :\Sigma \to (M,g)\) be an immersion of an oriented surface \(\Sigma\) into an oriented Riemannian \(4\)-manifold. The bundle of orientation compatible \(g\)-orthonormal coframes of \((M,g)\) is an \(\mathrm{SO}(4)\)-bundle \(\pi : F^+_g \to M\). The Grassmannian \(G^+_2(\mathbb{R}^4)\) of oriented \(2\)-planes in \(\mathbb{R}^4\) is a homogeneous space for the natural action of \(\mathrm{SO}(4)\) and the stabiliser subgroup is \(\mathrm{SO}(2)\times \mathrm{SO}(2)\). Consequently, the pullback bundle \(X^*F^+_g \to \Sigma\) admits a reduction \(F_X\subset X^*F^+_g\) with structure group \(\mathrm{SO}(2)\times \mathrm{SO}(2)\), where the fibre of \(F_X\) at \(p \in \Sigma\) consists of those coframes mapping the oriented tangent plane to \(\Sigma\) at \(X(p)\) to some fixed oriented \(2\)-plane in \(\mathbb{R}^4\), while preserving the orientation.

The second fundamental form of \(X\) is a quadratic form on \(T\Sigma\) with values in the rank two normal bundle of \(X\). Therefore, it is represented by a map \(F_X \to S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) which is equivariant with respect to some suitable representation of \(\mathrm{SO}(2)\times \mathrm{SO}(2)\) on \(S^2(\mathbb{R}_2)\times \mathbb{R}^2\). The relevant representation is defined by the rule \[\varrho(r_{\alpha},r_{\beta})(A)(x,y)=r_{-\beta}A(r_{\alpha}x,r_{\alpha}y), \quad x,y \in \mathbb{R}^2,\] where \(A \in S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) is a symmetric bilinear form on \(\mathbb{R}^2\) with values in \(\mathbb{R}^2\) and \(r_{\alpha},r_{\beta}\) denote counter-clockwise rotations in \(\mathbb{R}^2\) by the angle \(\alpha,\beta\), respectively. As usual, we decompose the \(\mathrm{SO}(2)\times \mathrm{SO}(2)\)-module \(S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) into irreducible pieces. This yields \[\varrho=\varsigma_{0,-1}\oplus\varsigma_{2,1}\oplus\varsigma_{2,-1},\] where for \((n,m) \in \mathbb{Z}^2\) the complex one-dimensional \(\mathrm{SO}(2)\times \mathrm{SO}(2)\)-representation \(\varsigma_{n,m}\) is defined by the rule \[\varsigma_{n,m}(r_{\alpha},r_{\beta})=\mathrm{e}^{\mathrm{i}(n \alpha+m\beta)}.\] Explicitly, the relevant projections \(S^2(\mathbb{R}_2)\otimes \mathbb{R}^2 \to \mathbb{C}\) are \[\begin{aligned} p_{0,-1}(A)&=\frac{1}{2}\left(A^1_{11}+A^1_{22}\right)+\frac{\mathrm{i}}{2}\left(A^2_{11}+A^2_{22}\right),\\ p_{2,-1}(A)&=\frac{1}{4}\left(A^1_{11}-A^1_{22}+2A^2_{12}\right)+\frac{\mathrm{i}}{4}\left(A^2_{11}-A^2_{22}-2A^1_{12}\right),\\ p_{2,1}(A)&=\frac{1}{4}\left(A^1_{22}-A^1_{11}+2A^2_{12}\right)+\frac{\mathrm{i}}{4}\left(A^2_{11}-A^2_{22}+2A^1_{12}\right) \end{aligned}\](4.1)–(4.3) and where we write \(A(e_i,e_j)=A^k_{ij}e_k\) with respect to the standard basis \((e_1,e_2)\) of \(\mathbb{R}^2\).

The canonical bundle \(K_{\Sigma}\) of \(\Sigma\) with respect to the complex structure induced by the metric \(X^*g\) and orientation is the bundle associated to the representation \(\varsigma_{1,0}\). Moreover, the conormal bundle, thought of as a complex line bundle, is the bundle \(N^*_X\) associated to the representation \(\varsigma_{0,1}\). Consequently, the second fundamental form of \(X\) defines a section \(H\) of the normal bundle which is the mean curvature vector of \(X\), as well as a quadratic differential \(Q_+\)with values in the conormal bundle and a quadratic differential \(Q_-\) with values in the normal bundle. Consequently, we obtain a quartic differential \(Q_+Q_-\) on \(\Sigma\) which turns out to be holomorphic, provided \(X\) is minimal and \(g\) has constant sectional curvature, see [11].

If we instead consider a split-signature oriented Riemannian \(4\)-manifold \((M,g)\), the bundle of orientation compatible \(g\)-orthonormal coframes of \((M,g)\) is an \(\mathrm{SO}(2,2)\)-bundle \(\pi : F^+_g \to M\). Here, as usual, we take \(\mathrm{SO}(2,2)\) to be the subgroup of \(\mathrm{SL}(4,\mathbb{R})\) stabilising the quadratic form \[q(x)=(x_1)^2+(x_2)^2-(y_1)^2-(y_2)^2,\] where \((x,y) \in \mathbb{R}^{2,2}\). Now the action of \(\mathrm{SO}(2,2)\) on the Grassmannian \(G^+_2(\mathbb{R}^{2,2})\) of oriented \(2\)-planes in \(\mathbb{R}^{2,2}\) is not transitive, it is however transitive on the open submanifolds of oriented timelike/spacelike \(2\)-planes. In both cases, the stabiliser subgroup is \(\mathrm{SO}(2)\times\mathrm{SO}(2)\) as well. Therefore, the submanifold theory of a timelike/spacelike oriented surface in an oriented split-signature Riemannian four manifold is entirely analogous to the Riemannian case. In particular, we also encounter the mean curvature vector \(H\) and the quadratic differentials \(Q_{\pm}\).

4.2 The mean curvature form

Knowing what to expect, we now carry out the submanifold theory of timelike/spacelike Lagrangian connections. Note however, that in addition to the split-signature metric \(h_{\nabla}\), we also have a symplectic form \(\Omega_{\nabla}\). The symplectic form allows to identify the conormal bundle to a Lagrangian spacelike/timelike immersion with the cotangent bundle of \(\Sigma\). In particular we may think of the mean curvature vector \(H\) as a \(1\)-form, the conormal-bundle valued quadratic differential \(Q_+\) as a cubic differential and the normal-bundle valued quadratic differential \(Q_-\) as a \((1,\! 0)\)-form.

The product \(P:=F\times \mathbb{R}_2\) is a principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle over \(T^*\Sigma\), where the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action is given by \((f,\xi)\cdot a=(a^{-1}\circ f,\xi a)\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\) and \((f,\xi) \in P\). We define two \(\mathbb{R}^2\)-valued \(1\)-forms on \(P\) \[\rho:=\frac{1}{2}\left(\psi^t+\omega\right)\quad \text{and}\quad \zeta:=\frac{1}{2}\left(\psi^t-\omega\right),\] so that the metric \(h_{\nabla}\) satisfies \[\pi^*h_{\nabla}=\rho^t\rho-\zeta^t\zeta=(\rho^1)^2+(\rho^2)^2-(\zeta^1)^2-(\zeta^2)^2.\] From the equivariance properties (2.7) and (2.8) of \(\psi\) and \(\omega\), we compute \[ R_a^*\begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\frac{1}{2}\begin{pmatrix} a^t+a^{-1} & a^t-a^{-1} \\ a^t-a^{-1} & a^t+a^{-1}\end{pmatrix}\begin{pmatrix} \rho \\ \zeta \end{pmatrix}.\](4.4)

Recall that the Lie algebra of the split-orthogonal group \(\mathrm{O}(2,2)\) consists of matrices of the form \[\begin{pmatrix} \mu & \nu \\ \nu^t & \vartheta \end{pmatrix}\] where \(\mu\) and \(\vartheta\) are skew-symmetric. Consequently, there exist unique \(\mathfrak{o}(2)\)-valued \(1\)-forms \(\mu,\vartheta\) on \(F\times \mathbb{R}_2\) and a unique \(\mathfrak{gl}(2,\mathbb{R})\)-valued \(1\)-form \(\nu\) on \(F\times \mathbb{R}_2\) such that \[ \mathrm{d}\begin{pmatrix} \rho \\ \zeta\end{pmatrix}=-\begin{pmatrix} \mu & \nu \\ \nu^t & \vartheta \end{pmatrix}\wedge \begin{pmatrix} \rho \\ \zeta\end{pmatrix}.\](4.5) In order to compute these connection forms we first remark that since the function \(S=(S_{ij})\) represents the (symmetric) Ricci tensor of \(\nabla\), there must exist unique real-valued functions \(S_{ijk}=S_{jik}\) on \(F\) so that \[\mathrm{d}S_{ij}=S_{ijk}\omega^k+S_{ik}\theta^k_j+S_{kj}\theta^k_i.\] Clearly, the function \((S_{ijk}) : F \to S^2(\mathbb{R}_2)\otimes \mathbb{R}_2\) represents \(\nabla\, \mathrm{Ric}(\nabla)\) with \(k\) being the derivative index.

Recall that the components of \(\psi\) and \(\omega\) – and hence equivalently the components of \(\rho\) and \(\zeta\) – span the \(1\)-forms on \(P\) that are semi-basic for the projection \(\pi : P \to T^*\Sigma\). In particular, if \(\epsilon\) is a \(1\)-form on \(T^*\Sigma\), then there exists a unique map \((e_1,e_2) : P \to \mathbb{R}_{2,2}\) so that \(\pi^*\epsilon=e_1\rho+e_2\zeta\). Since \(\pi^*\epsilon\) is invariant under the \(\mathrm{GL}^+(2,\mathbb{R})\) right action, the function \((e_1,e_2)\) satisfies the equivariance property determined by (4.4). Phrased differently, the cotangent bundle of \(T^*\Sigma\) is the bundle associated to \(\pi : P\to T^*\Sigma\) via the representation \(\varrho : \mathrm{GL}^+(2,\mathbb{R}) \to \mathbb{R}_{2,2}\) defined by the rule \[ \varrho(a)\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}=\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}\frac{1}{2}\begin{pmatrix} a^t+a^{-1} & a^t-a^{-1} \\ a^t-a^{-1} & a^t+a^{-1}\end{pmatrix}\](4.6) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\) and \((\xi_1,\xi_2)\) in \(\mathbb{R}_{2,2}\).

We will next use this fact to exhibit the conormal bundle of the immersion \(o : \Sigma \to T^*\Sigma\) as an associated bundle to a natural reduction of the pullback bundle \(o^*P\). Note that by construction, the pullback bundle \(o^*P \to \Sigma\) is just the frame bundle \(\upsilon : F \to \Sigma\) and that on \(o^*P\simeq F\) we have \(\psi^t=-S\omega\), thus \[\rho=\frac{1}{2}\left(\mathrm{I}_2-S\right)\omega \quad \text{and}\quad \zeta=-\frac{1}{2}\left(\mathrm{I}_2+S\right)\omega.\] If we assume that \(\nabla\) is timelike/spacelike, then the Ricci tensor of \(\nabla\) is positive/negative definite and hence the equations \(S=\pm \mathrm{I}_2\) define a reduction \(F_{\nabla} \to \Sigma\) with structure group \(\mathrm{SO}(2)\) whose basepoint projection we continue to denote by \(\upsilon\). Note that by construction, \(F_{\nabla} \to \Sigma\) is the bundle of orientation preserving orthonormal coframes of the induced metric \(g=\pm \mathrm{Ric}(\nabla)\). In particular, from (4.6) we see that the pullback bundle \(o^*(T^*(T^*\Sigma))\) is the bundle associated to \(\upsilon : F_{\nabla} \to \Sigma\) via the \(\mathrm{SO}(2)\)-representation on \(\mathbb{R}_{2,2}\) defined by the rule \[ \varrho(a)\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}=\begin{pmatrix} \xi_1 a^t & \xi_2 a^t\end{pmatrix}\](4.7) for all \(a \in \mathrm{SO}(2)\) and \((\xi_1,\xi_2)\) in \(\mathbb{R}_{2,2}\). Furthermore, on \(F_{\nabla}\) we obtain \(\psi=\mp \omega^t\) and hence \[ \begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\begin{pmatrix} 0\\ -\omega \end{pmatrix}\](4.8) in the timelike case and \[ \begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\begin{pmatrix} \omega\\ 0 \end{pmatrix}\](4.9) in the spacelike case. Recall that if \(\alpha\) is a \(1\)-form on \(\Sigma\) then there exists a unique \(\mathbb{R}_2\)-valued function \(a\) on \(F\) – and hence on \(F_{\nabla}\) as well – so that \(\upsilon^*\alpha=a\omega\). It follows as before that \(T^*\Sigma\) is the bundle associated to \(F_{\nabla}\) via the \(\mathrm{SO}(2)\)-representation defined by the rule \[ \varrho(a)(\xi)=\xi a^t\](4.10) for all \(a \in \mathrm{SO}(2)\) and \(\xi \in \mathbb{R}_2\). Using (4.7), (4.8) and (4.9), we see that the conormal bundle \[N^*_o:=o^*(T^*(T^*\Sigma))/T^*\Sigma\] of \(o\) is (isomorphic to) the bundle associated to \(F_{\nabla}\) via the representation (4.10) as well. We thus have an isomorphism \(N^*_o\simeq T^*\Sigma\) between the conormal bundle of the immersion \(o : \Sigma \to T^*\Sigma\) and the cotangent bundle \(T^*\Sigma\). Of course, the metric \(g\) on \(\Sigma\) provides an isomorphism \(T^*\Sigma\simeq T\Sigma\) and hence \(N_o\simeq T^*\Sigma\), where \(N_o\) denotes the normal bundle of \(o\). The second fundamental form of \(o\) is a quadratic form on \(T\Sigma\) with values in the normal bundle, thus here naturally a section of \(S^2(T^*\Sigma)\otimes T^*\Sigma\).

Denoting by \(S^{ij}\) the functions on \(F\) representing the inverse of the Ricci curvature of \(\nabla\), so that \(S^{ij}S_{jk}=\delta^i_k\), we thus have:

Recall that a (non-degenerate) submanifold of a (pseudo-)Riemannian manifold is called totally geodesic if its second fundamental form vanishes identically. We call a timelike/spacelike connection \(\nabla\) totally geodesic if \(o : \Sigma \to (T^*\Sigma,h_{\nabla})\) is a totally geodesic submanifold. We also get:

4.3 Minimality and the Liouville curvature

The minimality condition for a timelike/spacelike Lagrangian connection can also be expressed in terms of the Liouville curvature of \(\nabla\). To this end we decompose the structure equation³ \[ \mathrm{d}S_{ij}=S_{ijk}\omega^k+S_{ik}\theta^k_j+S_{kj}\theta^k_i\](4.13) into (we compute modulo \(\theta^i_j\)) \[\begin{aligned} \mathrm{d}S_{ij}&=\left(\frac{1}{3}(S_{ijk}+S_{ikj}+S_{jki})+\frac{2}{3}S_{ijk}-\frac{1}{3}(S_{ikj}+S_{jki})\right)\omega_k\\ &=\left(R_{ijk}+\frac{2}{3}L_{(i}\varepsilon_{j)k}\right)\omega^k, \end{aligned}\] where we define \[R_{ijk}=\frac{1}{3}\left(S_{ijk}+S_{ikj}+S_{jki}\right)\] and \[L_i=\frac{2}{3}\varepsilon^{jk}\left(S_{ijk}-\frac{1}{2}(S_{ikj}+S_{jki})\right).\]

Writing \(g=\pm \mathrm{Ric}(\nabla)\) for the induced metric, we define \(\beta \in \Omega^1(\Sigma)\) by \[\beta=\frac{3}{8}\operatorname{tr}_g \mathrm{Sym}\nabla g,\] where \(\mathrm{Sym} : \Gamma\left(T^*\Sigma\otimes S^2(T^*\Sigma)\right) \to \Gamma\left(S^3(T^*\Sigma)\right)\) denotes the natural projection. We have:

5 Minimal Lagrangian connections

We will next compute the structure equations of a timelike/spacelike minimal Lagrangian connection \(\nabla\). As before, we work on the orthonormal coframe bundle \(F_g\) of the induced metric \(g=\pm\mathrm{Ric}(\nabla)\). The submanifold theory discussed in §3 tells us that the second fundamental form of \(o : \Sigma \to (T^*\Sigma,h_{\nabla})\) is described in terms of a cubic differential and a \((1,\! 0)\)-form. Using the definition (4.2) of the \((1,\! 0)\)-form, the expression (4.11) for the second fundamental form and the minimality conditions (4.16), one easily computes that on \(F_g\) the \((1,\! 0)\)-form is represented by the complex-valued function \[b=\pm\frac{3}{8}\left((R_{111}+R_{221})-\mathrm{i}(R_{112}+R_{222})\right).\] Hence comparing with (4.15), we conclude that \(b=b_1-\mathrm{i}b_2\). Note that if we define the \((1,\! 0)\)-form \[\beta^{1,0}:=\beta+\mathrm{i}\star_g\beta,\] then we have \(\upsilon^*\beta^{1,0}=(b_1-\mathrm{i}b_2)(\omega_1+\mathrm{i}\omega_2)\), thus the \((1,\! 0)\)-form obtained from the normal bundle valued quadratic differential \(Q_-\) by using the symplectic form \(\Omega_{\nabla}\) is \(\beta^{1,0}\). Likewise, \(Q_+\) gives the cubic differential \(C\) on \(\Sigma\) which is represented on \(F_g\) by the complex-valued function \[c=\mp\frac{1}{8}\left(\left(R_{111}-3R_{122}\right)+\mathrm{i}\left(-3R_{112}+R_{222} \right)\right).\] From Lemma 2.1 and \[\upsilon^*\left(\mathrm{Sym}_0\mathrm{Ric}(\nabla)\right)=\left(R_{ijk}-\frac{3}{2}\delta_{(ij}R_{k)lm}\delta^{lm}\right)\omega_i\otimes\omega_j\otimes \omega_k\] we easily compute that the cubic differential \(C\) satisfies \(\operatorname{Re}(C)=\mp\frac{1}{2}\mathrm{Sym}_0\nabla g\), where the subscript \(0\) denotes the trace-free part with respect to \(g\).

The structure equations can now be summarised as follows:

Conversely, unravelling our computations backwards, we also get:

We immediately obtain:

Another consequence of the structure equation is:

6 The spherical case

The system of equations governing minimal Lagrangian connections are easy to analyse on the \(2\)-sphere \(S^2\):

7 The case of negative Euler-characteristic

Before we address the classification of minimal Lagrangian connections on compact surfaces of negative Euler characteristic, we observe that every projectively flat spacelike minimal Lagrangian connection defines a properly convex projective structure. Indeed, Labourie gave the following characterisation of properly convex projective manifolds:

We immediately obtain:

7.1 Classification

In Section 5 we have seen that a triple \((g,\beta,C)\) on an oriented surface \(\Sigma\) satisfying (5.1),(5.2),(5.3) uniquely determines a minimal Lagrangian connection on \(T\Sigma\). In this section we will show that in the case where \(\Sigma\) is compact and has negative Euler characteristic \(\chi(\Sigma)\), the conformal equivalence \([g]\) of \(g\) and the cubic differential \(C\) also uniquely determine \((g,\beta,C)\) and hence the connection, provided \(C\) does not vanish identically. In the case where \(C\) does vanish identically the connection is determined uniquely in terms of \([g]\) and \(\beta\).

We start by showing that there are no timelike minimal Lagrangian connections on a compact oriented surface of negative Euler-characteristic (the reader may also compare this with [17]).

Without loosing generality we henceforth assume that the torsion-free minimal Lagrangian connection \(\nabla\) on a compact oriented surface \(\Sigma\) with \(\chi(\Sigma)<0\) is spacelike. We will show that the triple \((g,\beta,C)\) defined by \(\nabla\) is uniquely determined in terms of \([g]\) and \((\beta,C)\).

Suppose \((g,\beta,C)\) with \(\beta\) closed satisfy \[K_g=-1+2\,|C|_g^2+\delta_g \beta.\] Let \(g_0\) denote the hyperbolic metric in \([g]\) and write \(g=\mathrm{e}^{2u}g_0\), so that \[\mathrm{e}^{-2u}(-1-\Delta_{g_0}u)=-1+2\mathrm{e}^{-6u}|C|_{g_0}^2+\mathrm{e}^{-2u}\delta_{g_0}\beta.\] We obtain \[-\Delta_{g_0}u=1+\delta_{g_0}\beta-\mathrm{e}^{2u}+2\mathrm{e}^{-4u}|C|^2_{g_0}.\] Omitting henceforth reference to \(g_0\) we will show:

Using the Hodge decomposition theorem it follows from the closedness of \(\beta\) that we may write \(\beta=\gamma+\mathrm{d}v\) for a real-valued function \(v \in C^{\infty}(\Sigma)\) and a unique harmonic \(1\)-form \(\gamma \in \Omega^1(\Sigma)\). Since \(\gamma\) is harmonic, it is co-closed, hence (7.1) becomes \[\Delta u=-1-\delta\mathrm{d}v+\mathrm{e}^{2u}-2\mathrm{e}^{-4u}|C|^2=-1+\Delta v+\mathrm{e}^{2u}-2\mathrm{e}^{-4u}|C|^2.\] Writing \(u^{\prime}:=u-v\), we obtain \[\Delta u^{\prime}=-1+e^{2(u^{\prime}+v)}-2\mathrm{e}^{-4(u^{\prime}+v)}|C|^2.\] Using the notation \(\kappa=-\mathrm{e}^{2v}<0\) and \(\tau=\mathrm{e}^{-4v}|C|^2\), as well as renaming \(u:=u^{\prime}\), we see that (7.4) follows from:

In order to prove this theorem we define an appropriate functional \(\mathcal{E}_{\kappa,\tau}\) on the Sobolev space \(W^{1,2}(\Sigma)\). As usual, we say a function \(u \in W^{1,2}(\Sigma)\) is a weak solution of (7.2) if for all \(\phi \in C^{\infty}(\Sigma)\) \[ 0=\int_{\Sigma}-\langle \mathrm{d}u,\mathrm{d}\phi\rangle+\left(1+\kappa\mathrm{e}^{2u}+2\tau\mathrm{e}^{-4u}\right)\phi\, dA.\](7.3) Note that this definition makes sense. Indeed, it follows from the Moser–Trudinger inequality that the exponential map sends the Sobolev space \(W^{1,2}(\Sigma)\) into \(L^{p}(\Sigma)\) for every \(p< \infty\), hence the right hand side of (7.3) is well defined.

Since \(\tau \geqslant 0\) we have \(\mathcal{E}_{\kappa,\tau}\geqslant \mathcal{E}_{\kappa,0}\) where here \(0\) stands for the zero-function. The functional \(\mathcal{E}_{\kappa,0}\) appears in the variational formulation of the equation for prescribed Gauss curvature \(\kappa\) of a metric \(g=\mathrm{e}^{2u}g_0\) on \(\Sigma\). In particular, \(\mathcal{E}_{\kappa,0}\) is well-known to be coercive and hence so is \(\mathcal{E}_{\kappa,\tau}\). In addition, we have:

We define the area of a timelike/spacelike connection to be the area of \(o(\Sigma)\subset (T^*\Sigma,h_{\nabla})\). We have:

Theorem 7.4 shows that the triple \((g,\beta,C)\) is uniquely determined in terms of the conformal equivalence class \([g]\), the cubic differential \(C\) and the \(1\)-form \(\beta\) on \(\Sigma\). Since \(C\) can locally be rescaled to be holomorphic, its zeros must be isolated and hence \(\beta\) is uniquely determined by \(C\) provided \(C\) does not vanish identically. Therefore, applying Corollary 5.3 shows:

7.2 Concluding remarks

References