6 Examples
Let \(M\) be a closed oriented surface equipped with a hyperbolic metric \(g_0\). Assume furthermore that the unit tangent bundle \(SM\) of \((M,g_0)\) admits an \(n\)-th root \(SM^{1/n}\), so that correspondingly we have an \(n\)-th root \(K^{1/n}\) of the canonical bundle \(K\) of \((M,g_0)\). Let \(m\) be a positive integer and write \(\ell=m/n\). We equip \(K^{1+\ell}\) with the holomorphic structure determined by \(g_0\), that is, in our previous notation, we choose \(\theta\equiv 0\). Suppose \(A\) is a holomorphic differential of fractional degree \(1+\ell\). Note that such differentials exist by the Riemann–Roch theorem. In order to obtain one of our Anosov flows, we must thus find a metric \(g\) in the conformal equivalence class of \(g_0\) so that \[K_g=-1+\ell |A|^2_g.\] Under a conformal change \(g_0 \mapsto e^{2u}g_0\) with \(u\in C^{\infty}(M)\), the norm \(|A|_{g_0}^2\) changes as \[|A|^2_{e^{2u}g_0}=e^{-2(1+\ell)u}|A|^2_{g_0}.\] We also have the identity \[K_{e^{2u}g_0}=e^{-2u}(-1-\Delta u)\] for the change of the Gauss curvature under conformal change. Here \(\Delta\) denotes the Laplace operator with respect to the hyperbolic metric. Writing \(g=e^{2u}g_0\), we thus obtain the PDE \[\Delta u=-1 +e^{2u}-\ell e^{-2\ell u}\alpha\] with \(\alpha:=|A|^2_{g_0}\). Since \(\alpha\geqslant 0\), this quasi-linear elliptic PDE admits a unique smooth solution which can be obtained by standard methods, see for instance [32]. Therefore, we obtain a solution to the vortex equations and an associated Anosov flow.
Recall that every closed oriented hyperbolic Riemann surface \((M,g_0)\) admits a Fuchsian model, which realises its unit tangent bundle \(SM\) as a quotient \(\Gamma\setminus \mathrm{PSL}(2,\mathbb{R})\), where \(\Gamma\subset \mathrm{PSL}(2,\mathbb{R})\) is a Fuchsian group, that is, a discrete torsion-free subgroup of \(\mathrm{PSL}(2,\mathbb{R})\). Therefore, we obtain a square root \(SM^{1/2}\simeq \tilde{\Gamma}\setminus \mathrm{SL}(2,\mathbb{R})\), where \(\tilde{\Gamma}\subset \mathrm{SL}(2,\mathbb{R})\) denotes the preimage of \(\Gamma\) under the \(2\)-fold cover \(\mathrm{SL}(2,\mathbb{R}) \to \mathrm{PSL}(2,\mathbb{R})\). Since the unit tangent bundles with respect to conformally equivalent metrics are isomorphic as principal \(\mathrm{SO}(2)\)-bundles, we also obtain a square root of the unit tangent bundle for every metric in the conformal equivalence class of \(g_0\). In particular, on every closed hyperbolic Riemann surface we obtain an Anosov flow on \(SM^{1/2}\) from a holomorphic differential \(A\) of fractional degree \(1+1/2=3/2\). These flows are topologically orbit equivalent to the lift of a constant curvature geodesic flow [16], but do not arise from the lift of a flow on \(SM\).