4 Applications
We consider now some special choices of \(\lambda\). To this end let \(\theta\) be a \(1\)-form on \(M\) which we may equivalently think of as a function \(\theta : SM \to \mathbb{R}\) satisfying \(VV\theta=-\theta\). For later use we record that the co-differential of \(\theta\) and its Hodge-star satisfy \[\tag{4.1} \pi^*\delta_g\theta=-(X\theta+HV\theta), \qquad \pi^*(\star_g\theta)=-V(\theta)\omega_1+\theta\omega_2.\] Moreover, let \(A\) be a differential of degree \(m\) on \(M\) with \(m\geqslant 2\). By this we mean a section of the \(m\)-th tensorial power of the canonical bundle \(K_{M}\) of \((M,g)\). Likewise, we may equivalently think of a differential \(A\) of degree \(m\) on \(M\) as a real-valued function \(a : SM \to \mathbb{R}\) satisfying \(VVa=-m^2a\), explicitly, we obtain \[\pi^*A=\left(Va/m+i a\right)\left(\omega_1+i \omega_2\right)^m,\] so that \[\tag{4.2} \pi^*|A|^2_g=(Va)^2/m^2+a^2\]
The thermostat flows we investigate are of the form \(\lambda=a-V\theta\). We will see next that they admit a dominated splitting provided a natural pair of equations is satisfied by the triple \((g,A,\theta)\). In order to derive these equations we first need a Lemma.
We have \[\tag{4.3} \overline{\partial}A=\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes A\quad\] iff \[\tag{4.4} 0=XVa-mHa-(m-1)(\theta Va-maV\theta).\]
Note that applying \(V\) we see that (4.4) is equivalent to \[\tag{4.5} 0=(1-m)\left(HVa+mXa-(m-1)\left(m\theta a+V(\theta)V(a)\right)\right).\]
Proof of Lemma 4.1. We use the complex notation \(\tilde{a}=Va/m+i a\) and \(\omega=\omega_1+i\omega_2\). Since \(VVa=-m^2a\), we compute that there exist unique complex-valued functions \(\tilde{a}^{\prime}\) and \(\tilde{a}^{\prime\prime}\) so that \[d\tilde{a}=\tilde{a}^{\prime}\omega+\tilde{a}^{\prime\prime}\overline{\omega}+im\tilde{a}\psi.\] In particular, we have \(\pi^*(\overline{\partial} A)=\tilde{a}^{\prime\prime}\overline{\omega}\otimes\omega^m\). Since \[\begin{aligned} da&=X(a)\omega_1+H(a)\omega_2+V(a)\psi,\\ d(Va)&=X(V(a))\omega_1+H(V(a))\omega_2-m^2a\psi, \end{aligned}\] we obtain \[\tilde{a}^{\prime\prime}=\frac{1}{2}\left(XVa/m-Ha\right)+\frac{i}{2}\left(HVa/m+Xa\right).\] We also have \[\pi^*\left(\theta-i\star_g\theta\right)=\left(\theta+i V\theta\right)\overline{\omega}.\] Hence (4.3) is equivalent to \[\tilde{a}^{\prime\prime}-\left(\frac{m-1}{2}\right)(\theta+iV\theta)(V(a)/m+i a)=0.\] Taking the real part gives (4.4).
Recall that a torsion-free connection on \(TM\) preserving a conformal structure \([g]\) is called a Weyl connection or conformal connection. More precisely, \(\nabla\) preserves \([g]\) if for some (and hence any) \(g\in[g]\), there exists a \(1\)-form \(\theta\), so that \[\nabla g=2\theta\otimes g.\]
. We could also consider the case \(\lambda=a-V\theta\) with \(a\) representing a differential of degree \(m=1\), that is, a \((1,\! 0)\)-form. We exclude this case since it corresponds to the case where \(A\) vanishes identically by defining \(\theta^{\prime}=Va\) and considering \(\lambda^{\prime}=-V(\theta^{\prime}-\theta)=\lambda\). Flows defined by \(\lambda=-V\theta=\) were studied previously under the name \(W\)-flows as they arise naturally by reparametrising the geodesics of a Weyl connection, see [41]. In particular in [41] it is proved that \(W\)-flows are Anosov provided \(K_g-\delta_g\theta<0\). A simple computation gives that \(\mathbb{K}=K_g-\delta_g \theta\) hence we recover [41] by applying Proposition 3.5. In particular, we see that if \(A\) is a holomorphic \(1\)-form and \(g\) satisfies \(K_g<0\), then the associated thermostat flow is Anosov.
We now want to apply Theorem 3.7 to the case \(\lambda=a-V\theta\) for some good choice of \(p\).
Suppose \(\lambda=a-V\theta\) and take \(p=Va/m+\theta\). Then \(\kappa_p\equiv -1\) if and only if the following two equations are identically satisfied \[\tag{4.6} K_g=-1+\delta_g\theta+(m-1)|A|^2_g\] and \[\tag{4.7} \overline{\partial}A=\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes A.\]
Proof. Taking \(p=Va/m+\theta\) gives \[\begin{aligned} \kappa_p&=\kappa+Fp+p(p-V\lambda)=K_g-H\lambda+\lambda^2+Fp+p(p-V\lambda)\\ &=K_g-Ha+HV\theta+a^2-2aV\theta+(V\theta)^2+(X+(a-V\theta)V)(Va/m+\theta)\\ &\phantom{=}+p(p-V\theta)\\ &=K_g+HV\theta+X\theta-(m-1)\left(a^2+(Va)/m^2\right)\\ &\phantom{=}+\left(XVa/m-Ha-(m-1)(\theta Va/m-aV\theta)\right)\\ &=K_g-\delta_g\theta-(m-1)|A|^2_g+\frac{1}{m}\left(XVa-mHa-(m-1)(\theta Va-maV\theta)\right),\end{aligned}\] where we have used (4.1), (4.2) and \(VVa=-m^2 a\) as well as \(VV\theta=-\theta\). Using Lemma 4.1 we see that \(\kappa_p\equiv -1\) provided (4.6) and (4.7) are identically satisfied. Conversely, suppose \(\kappa_p\equiv -1\). Since \(K_g-\delta_g\theta-(m-1)|A|^2_g\) is constant along the fibres of \(SM \to M\), we obtain \[0=V\kappa_p=\left(\frac{1-m}{m}\right)\Big(HVa+mXa-(m-1)\left(m\theta a+V(\theta)V(a)\right)\Big).\] Lemma 4.1 and Remark 4.2 therefore imply that (4.7) must hold. Hence we also identically have \[\kappa_p=-1=K_g-\delta_g\theta-(m-1)|A|^2_g,\] which is equivalent to (4.6).
Combining Theorem 3.7 and Lemma 4.5 we thus immediately obtain:
Let \((g,A,\theta)\) be a triple on \(M\) satisfying (4.6) and (4.7). Then the associated thermostat flow admits a dominated splitting.
We also observe:
Consider a pair \((g,A)\) with \(A\) holomorphic and \(K_g<0\). Then the associated thermostat flow has a dominated splitting. Moreover, for \(m=2\), the flow is Anosov.
Proof. The fact that there is a dominated splitting follows from \(\kappa_{p}<0\). For \(m=2\) we note that \[\kappa_{p}=K_g-|A|^2_g=K_g-a^{2}-(Va)^{2}/4.\] Thus \(\kappa_{p}+(Va)^{2}/4<0\) and the Anosov property follows from Theorem 3.7.
4.1 Parametrising thermostat flows arising from differentials
It turns out that the thermostat flows defined by triples \((g,A,\theta)\) satisfying (4.6) and (4.7) can be parametrised in terms of complex geometric data. For \(m\geqslant 2\) define the (smooth) complex line bundle \(L_m:=\Lambda^2(TM)^{(m-1)/2}\otimes {\mathbb C}\).
There exists a canonical bijection between the following sets:
the holomorphic line bundle structures on \(L_m\);
the \([g]\)-conformal connections on \(TM\).
Before we prove Lemma 4.8, we first recall some basic facts about conformal connections. Let us fix a Riemannian metric \(g \in [g]\). It follows from Koszul’s identity that the \([g]\)-conformal connections are of the form \[{}^{(g,\theta)}\nabla={}^g\nabla+g\otimes \theta^{\sharp}-\theta\otimes \mathrm{Id}-\mathrm{Id}\otimes \theta\] where \(\theta \in \Omega^1(M)\), \({}^g\nabla\) denotes the Levi-Civita connection of \(g\) and \(\theta^{\sharp}\) the \(g\)-dual vector field of \(\theta\). Moreover, for \(u \in C^{\infty}(M)\), we have [6] \[{}^{\exp(2u)g}\nabla={}^g\nabla-g\otimes{}^g\nabla u+du\otimes \mathrm{Id}+\mathrm{Id}\otimes du\] from which one easily computes \[{}^{(\exp(2u)g,\theta+du)}\nabla={}^{(g,\theta)}\nabla.\] Since \({}^{(g,\theta)}\nabla g=2\,\theta\otimes g\) and \({}^{(g,\theta)}\nabla \mathrm{e}^{2u}g=2\,(\theta+du)\otimes \mathrm{e}^{2u}g\), we conclude that the \([g]\)-conformal connections are in one-to-one correspondence with Weyl structures, where by a Weyl structure we mean an equivalence class \([g,\theta]\) subject to the equivalence relation \[(g,\theta)\sim (\hat{g},\hat{\theta})\quad \iff \quad \hat{g}=\mathrm{e}^{2u}g \;\; \text{and}\;\; \hat{\theta}=\theta+du\] for \(u \in C^{\infty}(M)\). For later usage we also record that the symmetric part of the Ricci curvature of \({}^{(g,\theta)}\nabla\) satisfies \[\mathrm{Sym}\,\mathrm{Ric}\left({}^{(g,\theta)}\nabla\right)=\left(K_g-\delta_g\theta\right)g.\]
Proof of Lemma 4.8. Let \(\overline{\partial}_{L_m} : \Gamma(M,L_m) \to \Omega^{0,1}(M,L_m)\) be a holomorphic line bundle structure on \(L_m\). Observe that \((\det g)^{-(m-1)/4}\) is a non-vanishing section of \(L_m\), hence \[(\det g)^{(m-1)/4}\otimes\overline{\partial}_{L_m}(\det g)^{-(m-1)/4}\] is a \((0,\! 1)\)-form on \(M\). Thus there exists a unique \(1\)-form \(\theta\) on \(M\) so that \[\overline{\partial}_{L_m}(\det g)^{-(m-1)/4}=-\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes (\det g)^{-(m-1)/4}.\] If we instead consider the metric \(\hat{g}=\mathrm{e}^{2u}g\) for \(u \in C^{\infty}(M)\), then we obtain \[\overline{\partial}_{L_m}(\det \hat{g})^{-(m-1)/4}=-\left(\frac{m-1}{2}\right)\left(\hat{\theta}-i\star_g\hat{\theta}\right)\otimes (\det \hat{g})^{-(m-1)/4}\] with \(\hat{\theta}=\theta+du\). It follows that \(\overline{\partial}_{L_m}\) defines a Weyl structure on \(M\). Moreover, if two holomorphic line bundle structures \(\overline{\partial}_{L_m}\) and \(\overline{\partial}^{\prime}_{L_m}\) on \(L_m\) determine the same Weyl structure \([g,\theta]\), then they satisfy \[\overline{\partial}_{L_m}(\det g)^{-(m-1)/4}=\overline{\partial}^{\prime}_{L_m}(\det g)^{-(m-1)/4}\] and hence also \(\overline{\partial}_{L_m}=\overline{\partial}^{\prime}_{L_m}\).
Conversely, let \({}^{(g,\theta)}\nabla\) be a \([g]\)-conformal connection, then \[{}^{(g,\theta)}\nabla \left(\det g\right)^{-(m-1)/4}=-\left(m-1\right)\theta\otimes \left(\det g\right)^{-(m-1)/4}.\] Extending \({}^{(g,\theta)}\nabla\) complex linearly, we obtain a connection on the complex line bundle \(L_m\) whose curvature form is (since \({\mbox {dim}}\,_{{\mathbb C}} M=1\)) an \(\mathrm{End}(L_m)\)-valued \((1,\! 1)\)-form on \(M\). Thus, standard results imply (c.f. [23]) that there exists a unique holomorphic line bundle structure \(\overline{\partial}_{L_m}\) on \(L_m\) so that \(\overline{\partial}_{L_m}={}^{(g,\theta)}\nabla^{(0,1)}\). Finally, we have \[\begin{aligned} {}^{(g,\theta)}\nabla^{(0,1)}\left(\det g\right)^{-(m-1)/4}&=-\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes (\det g)^{-(m-1)/4}\\ &=\overline{\partial}_{L_m}(\det g)^{-(m-1)/4}.\end{aligned}\] Therefore, the Weyl structure determined by \(\overline{\partial}_{L_m}\) is \([g,\theta]\), thus proving the claim.
Given a section \(P\) of \(L_{m}\otimes K_{M}^{m}\) we can define \[|P|_{g}^2:=|A|^{2}_{g}\] where \(A:=\left(\det g\right)^{(m-1)/4}\otimes P\). It is straightforward to check that the quadratic form \[\mathbb{P}:=|P|^{2}_{g}g\] only depends on \([g]\).
We now have:
Let \(m\geqslant 2\). On a compact oriented surface \(M\) with \(\chi(M)<0\) the following sets are in one-to-one correspondence:
the triples \((g,A,\theta)\) consisting of a Riemannian metric \(g\), a differential \(A\) of degree \(m\) and a \(1\)-form \(\theta\) such that \[K_g=-1+\delta_g\theta+(m-1)|A|^2_g\quad \text{and}\quad \overline{\partial} A=\left(\frac{m-1}{2}\right)\left(\theta-i\star\theta\right)\otimes A;\]
the triples \(([g],\overline{\partial}_{L_m},P)\) consisting of a conformal structure \([g]\), a holomorphic line bundle structure \(\overline{\partial}_{L_m}\) on \(L_m\) and a holomorphic section \(P\) of \(L_m\otimes K_M^m\) having the property that the symmetric part of the Ricci curvature of the conformal connection associated to \(\overline{\partial}_{L_m}\) plus \((1-m)\mathbb{P}\) is negative definite.
Proof. Suppose \((g,A,\theta)\) is a triple satisfying \[K_g=-1+\delta_g\theta+(m-1)|A|^2_g \quad \text{and}\quad \overline{\partial}A=\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes A.\] We equip \(L_m\) with the holomorphic line bundle structure induced by the conformal connection \({}^{(g,\theta)}\nabla\). Define \(P:=\left(\det g\right)^{-(m-1)/4}\otimes A\), then \(P\) is a holomorphic section of \(L_m\otimes K_M^m\). Indeed, we compute \[\begin{aligned} \overline{\partial}P&=\overline{\partial}_{L_m}\left((\det g)^{-(m-1)/4}\right)\otimes A+\left(\det g\right)^{-(m-1)/4}\otimes \overline{\partial}_{K_M}A\\ &=-\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes P+\left(\frac{m-1}{2}\right)\left(\theta-i\star_g\theta\right)\otimes P\\ &=0.\end{aligned}\] In addition, we observe that the symmetric part of the Ricci curvature of \({}^{(g,\theta)}\nabla\) satisfies \[\mathrm{Sym}\;\mathrm{Ric}\left({}^{(g,\theta)}\nabla\right)+(1-m)\mathbb{P}=\left(K_g-\delta_g\theta+(1-m)|A|_{g}^{2}\right)g=-g\] which is obviously negative definite. Clearly, the just described map from the first set of triples into the second set of triples is injective.
Conversely, suppose \(L_m\) is equipped with a holomorphic line bundle structure \(\overline{\partial}_{L_m}\) and let \(P\) be a holomorphic section of \(L_m\otimes K_M^m\). Assume furthermore that the symmetric part of the Ricci curvature of the conformal connection associated to \(\overline{\partial}_{L_m}\) plus \((1-m)\mathbb{P}\) is negative definite. We will next use these data to construct a triple \((g,A,\theta)\) solving the above equations. Let \(g_0 \in [g]\) denote the hyperbolic metric in the conformal equivalence class and define \[A_0:=\left(\det g_0\right)^{(m-1)/4}\otimes P.\] Note that \((\det g_0)^{(m-1)/4}\) is a non-vanishing section of \(L_m^{-1}\) and hence \(A_0\) is a section of \(K_M^m\). Since \(P\) is holomorphic it follows that there exists a unique \(1\)-form \(\theta_0\) on \(M\) such that \[\overline{\partial}A_0=\left(\frac{m-1}{2}\right)\left(\theta_0-i\star\theta_0\right)\otimes A_0.\] Now make the Ansatz \(g=\mathrm{e}^{2u}g_0\) for \(u \in C^{\infty}(M)\) and \(A=\left(\det g\right)^{(m-1)/4}\otimes P=A_0\mathrm{e}^{u(m-1)}\). Then \[\overline{\partial}A=\left(\frac{m-1}{2}\right)\left(\theta-i\star\theta\right)\otimes A,\] where \(\theta=\theta_0+du\). Since \[\tag{4.8} K_{\exp(2u)g}=\mathrm{e}^{-2u}\left(K_g-\Delta_g u\right),\] where \(\Delta_g=-\left(\delta_{g}d+d\delta_{g}\right)\), we obtain \[\mathrm{e}^{-2u}\left(-1-\Delta u\right)=-1+\mathrm{e}^{-2u}\delta\left(\theta_0+du\right)+(m-1)\mathrm{e}^{-2u}|A_0|^2,\] where now all norms and operators are with respect to \(g_0\). This simplifies to become an algebraic equation for \(u\) \[\mathrm{e}^{2u}-(m-1)|A_0|^2=1+\delta \theta_0.\] Clearly, this equation uniquely determines \(u\) provided \(1+\delta\theta_0+(m-1)|A_{0}|^{2}\) is positive. Note that this happens if and only if \[(-1-\delta\theta_0+(1-m)|A_{0}|^{2})g_0=\mathrm{Sym}\,\mathrm{Ric}\left({}^{(g_0,\theta_0)}\nabla\right)+(1-m)\mathbb{P}\] is negative definite, but \({}^{(g_0,\theta_0)}\nabla\) is just the conformal connection induced by \(\overline{\partial}_{L_m}\). Finally, by construction, the triple associated to \((g,A,\theta)\) is \(([g],\overline{\partial}_{L_m},P)\).
The W-Flows of Wojtkowski [41] are also covered by the thermostat flows defined by triples \((g,A,\theta)\) satisfying (4.6) and (4.7) in the case where the conformal connection \({}^{(g,\theta)}\nabla\) defining the W-flow has negative definite symmetric Ricci curvature, that is, satisfies \((K_g-\delta_g\theta)<0\). Indeed, suppose the pair \((g,\theta)\) satisfies \((K_g-\delta_g\theta)<0\). Let \(u=\frac{1}{2}\ln\left(\delta_g\theta-K_g\right)\) and consider \((\hat{g},\hat{\theta})=(\mathrm{e}^{2u}g,\theta+du)\). Then the pairs \((g,\theta)\) and \((\hat{g},\hat{\theta})\) define the same conformal connection and hence equivalent W-flows. Using (4.8) and the identity \(\delta_{\exp(2u)g}=\mathrm{e}^{-2u}\delta_g\) for the co-differential acting on \(1\)-forms, we compute \[K_{\hat{g}}-\delta_{\hat{g}}\hat{\theta}=\left(\frac{1}{\delta_g\theta-K_g}\right)\left(K_g-\Delta_g u\right)-\left(\frac{1}{\delta_g\theta-K_g}\right)\delta_g\left(\theta+du\right)=-1.\] Hence the triple \((\hat{g},0,\hat{\theta})\) satisfies (4.6) and (4.7). In particular, we see that the geodesic flow of metrics of negative Gauss curvature also fit into our family of flows.