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:
Let \((M,\mathfrak{p})\) be an oriented flat projective manifold. Then the following statements are equivalent:
\(\mathfrak{p}\) is properly convex;
there exists a connection \(\nabla \in \mathfrak{p}\) preserving a volume form and whose Ricci curvature is negative definite.
We immediately obtain:
Let \(\nabla\) be a projectively flat spacelike minimal Lagrangian connection on the oriented surface \(\Sigma\). Then \(\nabla\) defines a properly convex projective structure.
Proof. In Remark 4.10 we have seen that a minimal Lagrangian connection \(\nabla\) is projectively flat if and only if \(\beta\) vanishes identically. In the projectively flat case the connection \(1\)-form \(\theta\) of \(\nabla\) thus is (see (5.5)) \[\theta=\begin{pmatrix} 0 & -\varphi \\ \varphi & 0\end{pmatrix}+\begin{pmatrix} c_1\omega^1-c_2\omega^2 & -c_2\omega^1-c_1\omega^2\\ -c_2\omega^1-c_1\omega^2 & -c_1\omega^1+c_2 \omega^2\end{pmatrix}.\] In particular, the trace of \(\theta\) vanishes identically and hence \(\nabla\) preserves the volume form of \(g\). Since \(\mathrm{Ric}(\nabla)=-g\), the claim follows by applying Labourie’s result.
7.1 Classification
In 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]).
Suppose \(\nabla^{\prime}\) is a minimal Lagrangian connection on the compact oriented surface \(\Sigma\) satisfying \(\chi(\Sigma)<0\). Then \(\nabla^{\prime}\) is spacelike.
Proof. Suppose \(\nabla^{\prime}\) were timelike and let \(g=\mathrm{Ric}(\nabla^{\prime})\). Then we obtain \[\int_{\Sigma}\operatorname{tr}_g\mathrm{Ric}(\nabla^{\prime})dA_g=2\int_{\Sigma}dA_g=2\,\mathrm{Area}(\Sigma,g)\geqslant 0\] and hence Proposition 3.2 and Theorem 3.5 imply that \[4\pi\chi(\Sigma)=\sup_{\mathfrak{p} \in \mathfrak{P}(\Sigma)}\inf_{\nabla \in \mathfrak{p}} \int_{\Sigma}\operatorname{tr}_g \mathrm{Ric}(\nabla)dA_g\geqslant 0,\] a contradiction.
Without losing 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:
Let \((\Sigma,g_0)\) be a compact hyperbolic Riemann surface. Suppose \(\beta \in \Omega^1(\Sigma)\) is closed and \(C\) is a cubic differential on \(\Sigma\). Then the equation \[\tag{7.1} -\Delta u=1+\delta\beta-\mathrm{e}^{2u}+2\mathrm{e}^{-4u}|C|^2\] admits a unique solution \(u \in C^{\infty}(\Sigma)\).
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 (5.1) follows from:
Let \((\Sigma,g_0)\) be a compact hyperbolic Riemann surface. Suppose \(\kappa,\tau \in C^{\infty}(\Sigma)\) satisfy \(\kappa<0\) and \(\tau \geqslant 0\). Then the equation \[\tag{7.2} -\Delta u=1+\kappa\mathrm{e}^{2u}+2\tau\mathrm{e}^{-4u}\] admits a unique solution \(u \in C^{\infty}(\Sigma)\).
This theorem can also be proved using the technique of sub – and supersolutions, see [17]. Here we instead use techniques from the calculus of variations.
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 \(\phi \in C^{\infty}(\Sigma)\) \[\tag{7.3} 0=\int_{\Sigma}-\langle \mathrm{d}u,\mathrm{d}\phi\rangle+\left(1+\kappa\mathrm{e}^{2u}+2\tau\mathrm{e}^{-4u}\right)\phi\, dA.\] 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.
Suppose \(u \in W^{1,2}(\Sigma)\) is a critical point of the functional \[\mathcal{E}_{\kappa,\tau} : W^{1,2}(\Sigma) \to \overline{\mathbb{R}}, \quad u \mapsto \frac{1}{2}\int_{\Sigma}|\mathrm{d}u|^2-2u-\kappa\mathrm{e}^{2u}+\tau\mathrm{e}^{-4u}dA.\] Then \(u \in C^{\infty}(\Sigma)\) and \(u\) solves (7.2).
Proof. For \(u,v\in W^{1,2}(\Sigma)\) we define \(\gamma_{u,v}(t)=u+t v\) for \(t \in \mathbb{R}\). We consider the curve \(\Gamma_{u,v}=\mathcal{E}_{\kappa,\tau}\circ \gamma_{u,v} : \mathbb{R}\to \mathbb{R}\) so that \[\begin{aligned} \Gamma_{u,v}(t)=\frac{1}{2}\int_{\Sigma}|\mathrm{d}u|^2+2t\langle \mathrm{d}u, \mathrm{d}v\rangle+t^2|\mathrm{d}v|^2\\-2(u+t v)-\kappa\mathrm{e}^{2(u+tv)}+\tau \mathrm{e}^{-4(u+tv)}dA. \end{aligned}\] The curve \(\Gamma_{u,v}(t)\) is differentiable in \(t\) with derivative \[\frac{\mathrm{d}}{\mathrm{d}t} \Gamma_{u,v}(t)=\int_{\Sigma} \langle \mathrm{d}u,\mathrm{d}v\rangle+t |\mathrm{d}v|^2-v-v\kappa \mathrm{e}^{2(u+tv)}-2v\tau \mathrm{e}^{-4(u+tv)}dA.\] Note that this last expression is well-defined. Again, it follows from the Moser–Trudinger inequality that \(\mathrm{e}^{2(u+tv)} \in L^2(\Sigma)\) for all \(u,v \in W^{1,2}(\Sigma)\) and \(t \in \mathbb{R}\). Since \(W^{1,2}(\Sigma)\subset L^2(\Sigma)\) it follows that \(v\mathrm{e}^{2(u+tv)}\) is in \(L^1(\Sigma)\) by Hölder’s inequality and thus so is \(v\mathrm{e}^{-4(u+tv)}\). In particular, assuming that \(u\) is a critical point and setting \(t=0\) after differentiation gives \[0=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0} \Gamma_{u,v}(t)=\int_{\Sigma}\langle \mathrm{d}u,\mathrm{d}v\rangle-v-v\kappa \mathrm{e}^{2u}-2v\tau \mathrm{e}^{-4u}dA.\] Since \(C^{\infty}(\Sigma)\subset W^{1,2}(\Sigma)\) it follows that \(u\) is a weak solution of (7.2). Since the right hand side of (7.2) is in \(L^{p}(\Sigma)\) for all \(p<\infty\), it follows from the Caldéron-Zygmund inequality that \(u \in W^{2,p}(\Sigma)\) for any \(p<\infty\). Therefore, by the Sobolev embedding theorem, \(u\) is an element of the Hölder space \(C^{1,\alpha}(\Sigma)\) for any \(\alpha<1\). Since the right hand side of (7.2) is Hölder continuous in \(u\), it follows from Schauder theory that \(u \in C^{2}(\Sigma)\), so that \(u\) is a classical solution of (7.2). Iteration of the Schauder estimates then gives that \(u \in C^{\infty}(\Sigma)\).
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:
The functional \(\mathcal{E}_{\kappa,\tau}\) is strictly convex on \(W^{1,2}(\Sigma)\).
Proof. Let \(u,v \in W^{1,2}(\Sigma)\) be given. Using the notation of the previous lemma, we observe that \(\Gamma_{u,v}(t)\) is twice differentiable in \(t\) with derivative \[\tag{7.4} \frac{\mathrm{d}^2}{\mathrm{d}t^2}\Gamma_{u,v}(t)=\int_{\Sigma} |\mathrm{d}v|^2-2v^2\kappa\mathrm{e}^{2(u+tv)}+8v^2\tau\mathrm{e}^{-4(u+tv)}dA.\] Note again that by Sobolev embedding \(v^2 \in L^2(\Sigma)\) for \(v \in W^{1,2}(\Sigma)\) and that both \(\mathrm{e}^{2(u+tv)}\) and \(\mathrm{e}^{-4(u+tv)}\) are in \(L^2(\Sigma)\), hence the right hand side of the equation (7.4) is well-defined by Hölder’s inequality. In particular, computing the second variation gives \[\begin{aligned} \mathcal{E}^{\prime\prime}_{\kappa,\tau}(u)[v,v]&=\left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\right|_{t=0}\mathcal{E}_{\kappa,\tau}(u+tv)\\ &=\int_{\Sigma}|\mathrm{d}v|^2dA+2\int_{\Sigma}v^2(4\tau-\mathrm{e}^{6u}\kappa)\mathrm{e}^{-4u}dA\\ &\geqslant \Vert \mathrm{d}v \Vert^2_{L^2(\Sigma)},\end{aligned}\] where we have used that \(\tau\geqslant 0\) and \(\kappa<0\). Since for a non-zero constant function \(v\) we obviously have \(\mathcal{E}^{\prime\prime}_{\kappa,\tau}(u)[v,v]>0\) it follows that the quadratic form \(\mathcal{E}_{\kappa,\tau}^{\prime\prime}\) is positive definite on \(W^{1,2}(\Sigma)\). Hence, the claim is proved.
Proof of Theorem 7.5. We have shown that \(\mathcal{E}_{\kappa,\tau}\) is a continuous strictly convex coercive functional on the reflexive Banach space \(W^{1,2}(\Sigma)\), hence \(\mathcal{E}_{\kappa,\tau}\) attains a unique minimum on \(W^{1,2}(\Sigma)\), see for instance [34]. Since we know that the minimum is smooth, Theorem 7.5 is proved.
We define the area of a timelike/spacelike connection to be the area of \(o(\Sigma)\subset (T^*\Sigma,h_{\nabla})\). We have:
Let \(\nabla\) be a minimal Lagrangian connection on the compact oriented surface \(\Sigma\) with \(\chi(\Sigma)<0\). Then we have \[\mathrm{Area}(\nabla)=- 2\pi\chi(\Sigma)+2\Vert C\Vert_g^2.\]
Proof. We have seen that the Gauss curvature of the metric \(o^*h_{\nabla}=g=-\mathrm{Ric}(\nabla)\) defined by a minimal Lagrangian connection \(\nabla\) on \(\Sigma\) satisfies \[K_g=-1+2\,|C|_g^2+\delta_g \beta.\] Integrating against \(dA_g\) and using the Stokes and Gauss–Bonnet theorem gives \[2\pi\chi(\Sigma)=-\mathrm{Area}(\nabla)+2\Vert C\Vert^2_g,\] thus proving the claim.
An obvious consequence of Theorem 7.9 is the area inequality \[\tag{7.5} \mathrm{Area}(\nabla)\geqslant -2\pi \chi(\Sigma)\] holding for minimal Lagrangian connections. Recall that if \(\nabla\) is projectively flat, then the projective structure defined by \(\nabla\) is properly convex. Labourie [26] associated to every properly convex projective surface \((\Sigma,\mathfrak{p})\) a unique minimal mapping from the universal cover \(\tilde{\Sigma}\) to the symmetric space \(\mathrm{SL}(3,\mathbb{R})/\mathrm{SO}(3)\) which satisfies the very same area inequality, that is (7.5), see [27]. Moreover, it is shown in [27] that equality holds if and only if \(\mathfrak{p}\) is defined by the Levi-Civita connection of a hyperbolic metric.
We call a minimal Lagrangian connection \(\nabla\) area minimising if \(\nabla\) has area \(-2\pi\chi(\Sigma)\).
Theorem 7.9 shows that a minimal Lagrangian connection \(\nabla\) is area minimising if and only if the induced cubic differential vanishes identically. We have shown in Proposition 5.5 that in the projectively flat case – when \(\beta\) vanishes identically – this translates to \(\nabla\) being the Levi-Civita connection of a hyperbolic metric, a statement in agreement with [27].
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:
Let \(\Sigma\) be a compact oriented surface with \(\chi(\Sigma)<0\). Then we have:
there exists a one-to-one correspondence between area minimising Lagrangian connections on \(T\Sigma\) and pairs \(([g],\beta)\) consisting of a conformal structure \([g]\) and a closed \(1\)-form \(\beta\) on \(\Sigma\);
there exists a one-to-one correspondence between non-area minimising minimal Lagrangian connections on \(T\Sigma\) and pairs \(([g],C)\) consisting of a conformal structure \([g]\) and a non-trivial cubic differential \(C\) on \(\Sigma\) that satisfies \(\overline{\partial} C=\left(\beta-\mathrm{i}\star_g \beta\right)\otimes C\) for some closed \(1\)-form \(\beta\).
7.2 Concluding remarks
We have proved that on a compact oriented surface \(\Sigma\) of negative Euler characteristic we have a bijective correspondence between projectively flat spacelike minimal Lagrangian connections and pairs \(([g],C)\) consisting of a conformal structure \([g]\) and a holomorphic cubic differential \(C\) on \(\Sigma\). By the work of Labourie [26] and Loftin [30], the latter set is also in bijective correspondence with the properly convex projective structures on \(\Sigma\). Since by Corollary 7.2 every projectively flat spacelike minimal Lagrangian connection defines a properly convex projective structure, we conclude that every such projective structure arises from a unique projectively flat spacelike minimal Lagrangian connection.
Recall that the universal cover \(\tilde{\Sigma}\) of a properly convex projective surface \((\Sigma,\mathfrak{p})\) is a convex subset of \(\mathbb{RP}^2\). Pulling back the minimal Lagrangian connection \(\nabla \in \mathfrak{p}\) to the universal cover gives a section of \(A \to \tilde{\Sigma}\), where now, by the work of Libermann [28], the total space of the affine bundle \(A \to \tilde{\Sigma}\) is contained in the submanifold of the para-Kähler manifold \(A_0\subset \mathbb{RP}^2\otimes \mathbb{RP}^{2*}\) consisting of non-incident point-line pairs. In particular, we obtain a minimal Lagrangian immersion \(\tilde{\Sigma} \to A\), recovering the result of Hildebrand [22, 23] in the case of two dimensions. Therefore, using the result of Loftin [30], one should be able to show that every properly convex projective manifold arises from a minimal Lagrangian connection.
The case of the \(2\)-torus can be treated with similar techniques, except for the possible occurrence of Lorentzian minimal Lagrangian connections. This will be addressed elsewhere.
In higher dimensions, the class of totally geodesic Lagrangian connections contains the Levi-Civita connection of Einstein metrics of non-zero scalar curvature. We also refer the reader to [7, 21] for a study of Einstein metrics in projective geometry.
In [31], the author has introduced the notion of an extremal conformal structure for a projective manifold \((M,\mathfrak{p})\). In two-dimensions, the naive Einstein AH structures of [17] appear to provide examples of projective surfaces admitting an extremal conformal structure. This may be taken up in future work.