6 The cases \(m=2\) and \(m=3\)

In this section we consider the special cases of \(m=2,3\) and their peculiarities. These flows have appeared in different contexts and for different reasons and in this section we explain these features.

6.1 The case \(m=2\)

Consider a pair \((g,A)\) where \(A\) is a quadratic differential with \(\bar{\partial}A=0\) and \(K_g=-1+|A|^{2}_{g}\). By Theorem 5.1, the associated thermostat flow is Anosov. These flows have the distinctive feature that their weak bundles are of class \(C^{\infty}\). Indeed for this case \(p=V(a)/2\), \(\kappa_{p}=-1\) and equation (3.4) reduces to \[Fh+h^{2}-1=0.\] From this we clearly see that \(r^{u,s}=\pm 1+V(a)/2\) and hence the weak bundles \[\mathbb{R}F\oplus\mathbb{R}(H+r^{s,u}V)\] are smooth. This class of thermostats flows was first considered in [33], where the coupled vortex equations for \(m=2\) were derived assuming that the weak foliations were smooth. Theorem 4.6 in [15] asserts that a smooth Anosov flow on a closed 3-manifold with weak stable and unstable foliations of class \(C^{1,1}\), is smoothly orbit equivalent to a suspension or to a quasi-fuchsian flow as described in [14]. (In our case, since we are working with circles bundles the latter alternative holds.) A quasi-fuchsian flow \(\psi\) depends on a pair of points \(([g_1],[g_2])\) in Teichmüller space, has smooth weak stable foliation \(C^{\infty}\)-conjugate to the weak stable foliation of the constant curvature metric \(g_1\) and smooth weak unstable foliation \(C^{\infty}\)-conjugate to the weak unstable foliation of the constant curvature metric \(g_2\). Moreover, \(\psi\) preserves a volume form if and only if \([g_1]=[g_2]\). The analogous result on the thermostat side is provided by Theorem 5.5 which asserts that the thermostat flow preserves a volume form iff \(A=0\). It is an interesting question (first raised in [33]) to decide if the thermostat flows originating from the coupled vortex equations \(\bar{\partial}A=0\), \(K_g=-1+|A|^{2}_{g}\) describe all possible quasi-fuchsian flows \(\psi\).

6.2 The case \(m=3\)

Let now \((g,A,\theta)\) be a triple on \(M\) satisfying (4.6) and (4.7) with \(A\) being a cubic differential. The connection form of the Levi-Civita connection on the tangent bundle \(TM\) is \[\begin{pmatrix} 0 & -\psi \\ \psi & 0\end{pmatrix}.\] We define a \(1\)-form on \(SM\) with values in \(\mathfrak{gl}(2,\mathbb{R})\) \[\begin{gathered} \Upsilon=(\Upsilon^i_j)=\begin{pmatrix} 0 & -\psi \\ \psi & 0 \end{pmatrix}\\ +\begin{pmatrix} (V(a)/3-\theta)\omega_1-(a+V(\theta))\omega_2 & -(V(\theta)+a)\omega_1+(\theta-V(a)/3)\omega_2\\ (V(\theta)-a)\omega_1-(\theta+V(a)/3)\omega_2 & -(\theta+V(a)/3)\omega_1+(a-V(\theta))\omega_2 \end{pmatrix}.\end{gathered}\] It is a consequence of the equivariance properties \[VVa=-9 a, \quad VV\theta=-\theta, \quad L_{V}\omega_1=\omega_2, \quad \text{and}\quad L_{V}\omega_2=-\omega_1\] that the \(1\)-form \(\Upsilon\) is the connection \(1\)-form of a unique (torsion-free) connection \(\nabla\) on the tangent bundle \(TM\). Moreover, since the interior product \(i_F\Upsilon^2_1\) vanishes identically for \(\lambda=a-V\theta\), it follows that the geodesics of the connection \(\nabla\) can be reparametrised to agree with the projections to \(M\) of the orbits of the thermostat flow defined by \(\lambda\), see [32] for details. Moreover, if \(\theta\) is closed the connection \(\nabla\) admits an interpretation as a Lagrangian minimal surface, see [31]. If \(A\) is holomorphic so that \(\theta\) vanishes identically, then the connection \(\nabla\) defines a properly convex projective structure on \(M\), see the work of Labourie [25] and [30, 31]. This means that the universal cover \(\Omega\) of \(M\) is a properly convex open subset of the real projective plane \(\mathbb{RP}^2\) for which there exists a discrete group \(\Gamma\) of projective transformations which acts cocompactly on \(\Omega\) and so that \(M=\Omega/\Gamma\). Thus, \((\Omega,\Gamma)\) is a divisible convex set. Moreover, the segments of the projective lines \(\mathbb{RP}^1\) contained in \(\Omega\) project to \(M\) to agree with the (unparametrised) geodesics of \(\nabla\). The universal cover \(\Omega\) being a convex set, it is equipped with the Hilbert metric. The geodesic flow of the Hilbert metric descends to \(\mathbb{S}M\) and by a result of Benoist [3], is Anosov if and only if \(\Omega\) is strictly convex. In [3], it is also shown that a divisible convex set is strictly convex if and only if the group dividing it is word-hyperbolic. Since the fundamental group of a closed surface of negative Euler characteristic is word-hyperbolic, it thus follows from known results that the thermostat flow associated to a holomorphic cubic differential is a reparametrisation of an Anosov flow. However, since the Anosov property is invariant under reparametrisation of the flow, we conclude that the thermostat flow associated to a holomorphic cubic differential is Anosov, which is the statement of our Theorem 5.1 for the special case \(m=3\).