1 Introduction

We introduce a new family of flows on the unit tangent bundle \(SM\) of a closed oriented Riemannian \(2\)-manifold \((M,g)\) of negative Euler characteristic. The flows are (generalised) thermostat flows and are generated by \(C^{\infty}\) vector fields of the form \(F:=X+(a-V\theta)V\), where \(X,V\) denote the geodesic and vertical vector fields on \(SM\), \(\theta\) is a \(1\)-form on \(M\) – thought of as a real-valued function on \(SM\) – and \(a\) represents a differential \(A\) of degree \(m\geqslant 2\) on \(M\). The triple \((g,A,\theta)\) determining the flow is subject to the equations \[\tag{1.1} K_g=-1+\delta_g\theta+(m-1)|A|^2_g\quad \text{and}\quad \overline{\partial}A=\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes A,\] where \(i=\sqrt{-1}\) and where \(K_g\) denotes the Gauss–curvature, \(\delta_g\) the co-differential and \(\star_g\) the Hodge-star with respect to \(g\) and the orientation. The case \(m=3\) of these equations appeared previously in [31] (assuming \(\theta\) is closed), where it is related to certain torsion-free connections on \(TM\) which admit an interpretation as Lagrangian minimal surfaces. Here we prove that our flows admit a dominated splitting and moreover, that this family of flows admits a parametrisation in terms of holomorphic data. Indeed, we show that a triple \((g,A,\theta)\) satisfying the equations (1.1) determines a holomorphic line bundle structure on the smooth complex line bundle \(L_m:=\Lambda^2(TM)^{(m-1)/2}\otimes {\mathbb C}\), so that the “weighted differential” \(P=\left(\det g\right)^{-(m-1)/4}\otimes A\) is a holomorphic section of \(L_m\otimes K_M^m\) and such that a certain negative curvature condition holds. Here \(K_M\) denotes the canonical bundle of \((M,g)\). Conversely, given a closed hyperbolic Riemann surface \((M,[g])\), a holomorphic line bundle structure on \(L_m\) and a holomorphic section \(P\) of \(L_m\otimes K^m_{M}\) satisfying a certain negative curvature condition, we construct a triple \((g,A,\theta)\) solving (1.1) and hence one of our flows, by using the uniformisation theorem and by solving an algebraic equation only.

In [41], Wojtkowski introduced W-flows by suitably reparametrising the geodesics of a Weyl connection (or conformal connection). We show that the case where \(A\) vanishes identically corresponds to W-flows associated to conformal connections on the tangent bundle of a surface that have negative definite symmetrised Ricci curvature. In particular, we recover [41], by showing that the flow associated to a triple \((g,0,\theta)\) solving (1.1) is Anosov. This is achieved by providing sufficiency conditions for a general thermostat flow to admit a dominated splitting and to have the Anosov property, see Proposition 3.5 and Theorem 3.7.

We then turn to the case where \(\theta\) vanishes identically, so that \(A\) is holomorphic, hence we have \[\tag{1.2} K_g=-1+(m-1)|A|^2_g \quad \text{and} \quad \overline{\partial} A=0.\] Note that applying standard quasi-linear elliptic PDE techniques we obtain a unique solution \(g\) to (1.2) for every holomorphic differential \(A\) on \((M,[g])\), see Remark 5.3. The equations (1.2) admit an interpretation as coupled vortex equations, see in particular [10]. The case \(m=2\) was considered in [33] in the context of Anosov thermostats admitting smooth weak bundles (see 6 for more details). In the case \(m=3\), the first equation is known as Wang’s equation in the affine sphere literature. In [38], Wang related its solution to a complete hyperbolic affine \(2\)-sphere in \(\mathbb{R}^3\), in particular \(g\) is known as the Blaschke metric. Moreover, for \(m=3\), a pair \((g,A)\) on \(M\) solving (1.2) defines a properly convex projective structure on \(M\) and hence turns \(M\) into a properly convex projective surface, see [25] and [29]. The universal cover \(\Omega\) of a properly convex projective surface of negative Euler characteristic is a strictly and properly convex domain in the projective plane \(\mathbb{RP}^2\) which admits a cocompact action by a group \(\Gamma\) of projective transformations. Consequently, we obtain a (two-dimensional) divisible convex set. Since \(\Omega\) is convex, it is equipped with the Hilbert metric and moreover, the Hilbert metric descends to define a Finsler metric on the quotient surface \(M\simeq \Omega/\Gamma\), see in particular [21] for a nice survey of these ideas. We observe that the geodesic flow of the Finsler metric is a \(C^1\) reparametrisation of the flow we associate to the pair \((g,A)\). Benoist has shown [3] that if \((\Omega,\Gamma)\) is a divisible convex set (not necessarily two-dimensional), then the geodesic flow of the Finsler metric \(F\) induced on \(\Omega/\Gamma\) – henceforth just called the Hilbert geodesic flow – is Anosov if and only if \(\Omega\) is strictly convex. Since the Anosov property is invariant under reparametrisation, we may ask if the thermostat flow associated to a pair \((g,A)\) solving (1.2) is Anosov for all \(m\geqslant 2\). This is indeed the case, we obtain:

The hyperbolicity properties of thermostats satisfying (1.1) are not apparent. To expose them, we first conjugate the derivative cocycle to another one in which we can see the effect of equations (1.1). This conjugation requires a careful choice of gauge, but once that is established, standard methods using quadratic forms give rise to a dominated splitting. To upgrade this dominated splitting to hyperbolic as in the case of Theorem 5.1 requires an additional ingredient in the form of Lemma 5.2 below which asserts that \(K_{g}<0\); this gives control on the potentially problematic size of \(A\).

In the same way as geodesic flows are paradigms of conservative systems, thermostats may be seen as paradigms of dissipative systems. The special case of Gaussian thermostats (\(a=0\)) has provided interesting models in nonequilibrium statistical mechanics [11, 12, 35]. The next theorem shows that Anosov thermostat flows determined by the coupled vortex equations are indeed dissipative except when \(A=0\).

We remark that due to a theorem of Ghys [13] Anosov thermostat flows are Hölder orbit equivalent to the geodesic flow of (any) negatively curved metric of \(M\) and hence transitive (to be precise, [13] establishes a topological equivalence and the Hölder orbit equivalence follows from [20]).

In [3], Benoist also observes that the regularity of the weak foliations of the Hilbert geodesic flow coincides with the regularity of the boundary of the divisible convex set \((\Omega,\Gamma)\). By a result of Benzécri [5], the boundary has regularity \(C^2\) if and only if \(\Omega\) is an ellipsoid, in which case the induced Finsler metric is Riemannian and hyperbolic. Hence one might speculate that if a solution to the coupled vortex equations (1.2) gives rise to an Anosov flow having a weak foliation of regularity \(C^2\), then \(A\) vanishes identically. While we cannot prove this in general, we use Theorem 1.2 to resolve the odd case:

The orbits of our flow – when projected to the surface \(M\) – define what is known as a path geometry on \(M\), that is, a prescription of a path on \(M\) for every direction in each tangent space. In the case where \(A\) vanishes identically the paths are the geodesics of a hyperbolic metric and in the case where \(m=3\) the paths are the geodesics of a properly convex projective structure. In both cases, the path geometry is flat, by which we mean it is locally equivalent to the path geometry of great circles on the \(2\)-sphere. In the final section of the article we show:

Holomorphic differentials appear naturally in higher Teichmüller theory and here we briefly provide some context for our results while referring the reader to the recent survey [39] by Wienhard for a nice introduction to this currently very active research topic. Generalizing Teichmüller space, Hitchin [18] identified a connected component \(\mathcal{H}(M,G)\) – nowadays called the Hitchin component – in the representation variety \(\mathrm{Hom}\left(\pi_1M,G\right)/G\), where \(M\) is a connected closed oriented surface of negative Euler characteristic and \(G\) a real split Lie group. Fixing a conformal structure \([g]\) on \(M\), Hitchin used the theory of Higgs bundles [19] to provide a parametrisation of \(\mathcal{H}(M,G)\) in terms of holomorphic differentials on \((M,[g])\). While Hitchin’s parametrisation of \(\mathcal{H}(M,G)\) relies on the choice of an arbitrary conformal structure \([g]\) on \(M\), Labourie [26] was recently able to construct a canonical parametrisation of \(\mathcal{H}(M,G)\) in the case where \(G\) is \(\mathrm{PSL}(3,\mathbb{R})\), \(\mathrm{PSp}(4,\mathbb{R})\) or the split form \(\mathrm{G}_{2,0}\) of the exceptional group \(\mathrm{G}_2\) (see also [25] and [29] for the case \(G=\mathrm{PSL}(3,\mathbb{R})\)). More precisely, Labourie obtains a mapping class group equivariant identification of \(\mathcal{H}(M,G)\) with the fibre bundle over Teichmüller space whose fibre at \(J\) is \(H^0(M,K_{M,J}^3)\), \(H^0(M,K_{M,J}^4)\) and \(H^0(M,K_{M,J}^6)\) respectively. By the work of Goldman [17] and Choi–Goldman [8] the component \(\mathcal{H}(M,\mathrm{PSL}(3,\mathbb{R}))\) consists of (conjugacy classes of) monodromy representations of properly convex projective structures on \(M\) and this together with the work of Labourie [25, 26] and Loftin [29] yields the aforementioned description of properly convex projective structures in terms of pairs \(([g],A)\) with \(A\) a holomorphic cubic differential.

Using the equivariant flag curve of Labourie [24], Potrie–Sambarino [34] associate several Anosov flows to every representation \(\rho\) in a certain neighbourhood of the Fuchsian locus in \(\mathcal{H}(M,\mathrm{PSL}(n,\mathbb{R}))\), \(n\geqslant 4\). In particular, using the canonical embeddings \[\mathcal{H}(M,\mathrm{PSp}(4,\mathbb{R}))\subset \mathcal{H}(M,\mathrm{PSL}(4,\mathbb{R})) \quad \text{and} \quad \mathcal{H}(M,\mathrm{G}_{2,0})\subset \mathcal{H}(M,\mathrm{PSL}(7,\mathbb{R})),\] the work of Labourie [24, 26] and Potrie–Sambarino [34] yields examples of Anosov flows for certain quartic and sixtic holomorphic differential on \((M,[g])\). It would be interesting to know how these flows relate to the flows introduced here. We plan to investigate this in future work.

Acknowledgements

The authors are grateful to Nigel Hitchin, Rafael Potrie and Andy Sanders for helpful conversations and the anonymous referee for her/his careful reading and many useful suggestions. GPP was partially funded by EPSRC grant EP/M023842/1.