1 Introduction

A projective structure on a smooth manifold \(M\) is an equivalence class \(\mathfrak{p}\) of torsion-free connections on its tangent bundle \(TM\), where two such connections are declared to be projectively equivalent if they share the same unparametrised geodesics. The set of torsion-free connections on \(TM\) is an affine space modelled on the sections of \(S^2(T^*M)\otimes TM\). By a classical result of Cartan, Eisenhart, Weyl (see [23] for a modern reference), two connections are projectively equivalent if and only if their difference is pure trace. In particular, it follows from the representation theory of \(\mathrm{GL}(2,\mathbb{R})\) that a projective structure on a surface \(M\) is a section of a natural affine bundle of rank \(4\) whose associated vector bundle is canonically isomorphic to \(V=S^3(T^*M)\otimes \Lambda^2(TM)\). Choosing an orientation and Riemannian metric \(g\) on \(M\), the bundle \(V\) decomposes into irreducible \(\mathrm{SO}(2)\)-bundles \(V\simeq T^*M\oplus S^3_0(T^*M)\), where the latter summand denotes the totally symmetric \((0,\! 3)\) tensors on \(M\) that are trace-free with respect to \(g\), or equivalently, the cubic differentials with respect to the complex structure \(J\) induced by \(g\) and the orientation. In other words, fixing an orientation and Riemannian metric \(g\) on \(M\), a projective structure \(\mathfrak{p}\) may be encoded in terms of a unique triple \((g,A,\theta)\), where \(A\) is a cubic differential – and \(\theta\) a \(1\)-form on \(M\). A conformal change of the metric \(g \mapsto \mathrm{e}^{2u}g\) corresponds to a change \[(g,A,\theta)\mapsto (\mathrm{e}^{2u}g,\mathrm{e}^{2u}A,\theta+du).\] Consequently, the section \(\Phi=A/d\sigma\) of \(K^2\otimes\overline{K^*}\) does only depend on the complex structure \(J\). Here \(d\sigma\) denotes the area form of \(g\) and \(K\) the canonical bundle of \(M\). In addition, we obtain a connection \(\mathrm{D}\) on the anti-canonical bundle \(K^*\) inducing the complex structure by taking the Chern connection with respect to \(g\) and by subtracting twice the \((1,\! 0)\)-part of \(\theta\). Again, the connection \(\mathrm{D}\) does only depend on \(J\). Fixing a complex structure \(J\) on \(M\) thus encodes a given projective structure \(\mathfrak{p}\) in terms of a unique pair \((\mathrm{D},\Phi)\).

There are two special cases of particular interest. Firstly, we can find a complex structure \(J\) so that \(\mathrm{D}\) is the Chern connection of a metric in the conformal class determined by \(J\). This amounts to finding a complex structure for which \(\theta\) is exact. Secondly, we can find a complex structure \(J\) so that \(\Phi\) vanishes identically. This turns out to be equivalent to \(\mathfrak{p}\) containing a Weyl connection for the conformal structure \([g]\) determined by \(J\), that is, a torsion-free connection on \(TM\) whose parallel transport maps are angle preserving with respect to \([g]\).

In [14], it is shown that a two-dimensional projective structure \(\mathfrak{p}\) does locally always contain a Weyl connection and moreover, finding the Weyl connection turns out to be equivalent to finding a holomorphic curve into a certain complex surface \(Z\) fibering over \(M\). Here we use this observation to rephrase the problem in terms of a non-linear PDE for a Beltrami differential. More precisely, we think of \(\mathfrak{p}\) as being given on a Riemann surface \((M,J)\) in terms of \((\mathrm{D},\Phi)\). We show (see Proposition 4.4) that \(\mathfrak{p}\) contains a Weyl connection with respect to the complex structure defined by the Beltrami differential \(\mu\) on \((M,J)\) if and only if \[\tag{1.1} \mathrm{D}^{\prime\prime}\mu-\mu\,\mathrm{D}^{\prime}\mu=\Phi\mu^3+\overline{\Phi},\] where \(\mathrm{D}^{\prime}\) and \(\mathrm{D}^{\prime\prime}\) denote the \((1,\! 0)\) – and \((0,\! 1)\)-part of \(\mathrm{D}\). Since every two-dimensional projective structure locally contains a Weyl connection, the above PDE for the Beltrami differential \(\mu\) can locally always be solved. Moreover, on the \(2\)-sphere every solution \(\mu\) lies in a complex \(5\)-manifold of solutions, whereas on a closed surface of negative Euler characteristic the solution is unique, provided it exists, see [15] (and Corollary 4.6 below).

Here we address the problem of finding a projective structure \(\mathfrak{p}\) for which the above PDE has no global solution. Naturally, one might start by looking at projective structures \(\mathfrak{p}\) at “the other end”, that is, those that arise from pairs \((\mathrm{D},\Phi)\) where \(\mathrm{D}\) is the Chern connection of a conformal metric, or equivalently, those for which there exists a metric \(g\) so that \(\mathfrak{p}\) is encoded in terms of the triple \((g,A,0)\). This class of projective structures includes the so-called properly convex projective structures. A projective surface \((M,\mathfrak{p})\) is called properly convex if it arises as a quotient of a properly convex open set \(\Omega\subset\mathbb{RP}^2\) by a free and cocompact action of a group \(\Gamma\subset \mathrm{SL}(3,\mathbb{R})\) of projective transformations. In particular, using the Beltrami–Klein model of two-dimensional hyperbolic geometry, it follows that every closed hyperbolic Riemann surface is a properly convex projective surface. Motivated by Hitchin’s generalisation of Teichmüller space [9], Labourie [11] and Loftin [12] have shown independently that on a closed oriented surface \(M\) of negative Euler characteristic every properly convex projective structure arises from a unique pair \((g,A,0)\), where \(g\) and \(A\) are subject to the equations \[K_g=-1+2|A|^2_g \quad \text{and}\quad \overline{\partial} A=0.\] Using quasilinear elliptic PDE techniques, C.P. Wang previously showed [24] (see also [6]) that the metric \(g\) is uniquely determined in terms of \(([g],A)\) by the equation for the Gauss curvature \(K_g\) of \(g\). Consequently, Labourie, Loftin conclude that on \(M\) the properly convex projective structures are in bijective correspondence with pairs \(([g],A)\) consisting of a conformal structure and a cubic holomorphic differential.

Naturally one might speculate that (1.1) does not admit a global solution for a properly convex projective structure \(\mathfrak{p}\) unless \(A\) vanishes identically, in which case \(\mathfrak{p}\) is hyperbolic. This is indeed the case:

This corollary is an application of the more general vanishing Theorem 6.1 (see below) whose proof makes use of a remarkable \(L^2\)-energy identity. This energy identity – known for geodesic flows as Pestov’s identity – is ubiquitous when solving uniqueness problems for X-ray transforms, including tensor tomography. To make the bridge between (1.1) and this circle of ideas, it is necessary to recast the non-linear PDE in dynamical terms as a transport problem. Given a projective structure \(\mathfrak{p}\) captured by the triple \((g,A,\theta)\) we associate a dynamical system on the unit tangent bundle \(\pi:SM\to M\) of \(g\) as follows. We consider a vector field of the form \(F=X+(a-V\theta) V\), where \(X,V\) denote the geodesic – and vertical vector field of \(SM\), \(a\in C^{\infty}(SM,\mathbb{R})\) represents the cubic differential \(A\) (essentially its imaginary part) and where we think of \(\theta\) as a function on \(SM\). The flow of the vector field \(F\) is a thermostat (see 3 below for more details) and it has the property that its orbits project to \(M\) as unparametrised geodesics of \(\mathfrak{p}\). We show that (1.1) is equivalent to the transport equation (see Corollary 5.6) \[\tag{1.2} Fu=Va+\beta\] on \(SM\), where the real-valued function \(u\) encodes a conformal metric of the sought after complex structure \(\hat{J}\) and \(\beta\) is a \(1\)-form on \(M\), again thought of as a function on \(SM\). Explicitly \[u=\frac{3}{2}\log\left(\frac{p}{(pq-r^2)^{2/3}}\right),\] where \(p,q,r\) are given in terms of a \(\hat{J}\)-conformal metric \(\hat{g}\) and the complex structure \(J\) of \((M,g)\) by \[p(x,v)=\hat{g}(v,v), \qquad r(x,v)=\hat{g}(v,Jv), \quad\text{and}\quad q(x,v)=\hat{g}(Jv,Jv).\] The right hand side in (1.2) has degree 3 in the velocities and the dynamics of \(F\) is Anosov when \(\mathfrak{p}\) is a properly convex projective structure [18], hence it is natural to think that techniques from tensor tomography might work. Regular tensor tomography involves the geodesic vector field \(X\) and the typical question at the level of the transport equation is the following: if \(Xu=f\) where \(f\) has degree \(m\) in the velocities, is it true that \(u\) has degree \(m-1\) in the velocities? The case \(m=2\) is perhaps the most important and it is at the core of spectral rigidity of negatively curved manifolds and Anosov surfaces [4, 7, 21]. Thermostats introduce new challenges, however we are able to successfully use a general \(L^2\) energy identity developed in [10] (following earlier results for geodesic flows in [22]) together with ideas in [18] to show that if equation (1.2) holds then \(a=0\) and \(\beta\) is exact. Our vanishing Theorem 6.1 is actually rather general and it applies to a class of projective structures considerably larger than properly convex projective structures, see Corollary 6.4 below.

For the case of surfaces with boundary a full solution to the tensor tomography problem was given in [20]; the solution was inspired by the proof of the Kodaira vanishing theorem in Complex Geometry. In the present paper, we go in the opposite direction, we import ideas from geometric inverse problems, to solve an existence question for a non-linear PDE in Complex Geometry. These connections were not anticipated, and it is natural to wonder if they are manifestations of something deeper.

Acknowledgements

The authors are grateful to Nigel Hitchin for helpful conversations. A part of the research for this article was carried out while TM was visiting FIM at ETH Zürich. TM thanks FIM for its hospitality. TM was partially funded by the priority programme SPP 2026 “Geometry at Infinity” of DFG. GPP was partially supported by EPSRC grant EP/R001898/1.