I am currently studying hyperbolic metrics on surfaces right now, and want to understand the construction of the Liouville Current for a metric $\varphi$ on my (orientable, compact) surface $S$. In particular, I am interested in when $\varphi$ has constant negative curvature $-1$. I have seen a couple of such constructions that I am able to follow, but I do not understand how any of them can be dependent on $\varphi$.
The simplest construction I have seen is via the ‘cross ratio’ of geodesics. Here is my understanding of it:
We interpret the space of geodesics $\mathcal{G}(S)$ on the surface $S$ by lifting each geodesic to the universal cover $\mathbb{H}^2$ and identifying a geodesic with its two endpoints at infinity. That way we can view $\mathcal{G}(S)$ as being $(S^1_{\infty} \times S^1_{\infty} - \Delta)/\sim$, where $\Delta$ is the diagonal of $S^1_{\infty} \times S^1_{\infty}$ and $\sim$ is the equivalence relation $(x,y) \sim (y,x)$.
Now pick four distinct points $a$, $b$, $c$, $d$ on the boundary of the Poincaré disk, and suppose I would like to measure the set of geodesics that have one point-at-infinity between $a$ and $b$ and another point-at-infinity between $c$ and $d$. Then letting $a_i$, $b_i$, $c_i$, and $d_i$ be sequences of points in $\mathbb{H}^2$ that converge to $a$, $b$, $c$, and $d$ respectively, we define the Liouville current measure of this set to be:
$$L_{\varphi}([a,b] \times [c,d]) = |C_{\varphi}(a,b,c,d)| = \frac{1}{2}\lim\limits_{i \to \infty} (d(a_i,d_i) + d(b_i, c_i) - d(a_i, c_i) - d(b_i, d_i)),$$
Where $d(\cdot, \cdot)$ is the distance function on $\mathbb{H}^2$. This defines our measure on closed rectangles in this space of geodesics, and so it can be extended to a borel measure on this space, which is the Liouville current $L_{\varphi}$.
My confusion is thus:
Where in this definition do we have dependence on $\varphi$? The sources I’ve looked at claim that the Liouville current is dependent on the metric. However for constant curvature hyperbolic surfaces their universal covers are all $\mathbb{H}^2$, i.e. they are isometric. And as we use the metric on $\mathbb{H}^2$ to define the cross-ratio function $C_{\varphi}$ it’s unclear why the measure of a set of geodesics would change if we began with a different metric $\varphi’$. What am I missing here?
The quantity in your post labelled $L_\phi([a,b] \times [c,d])$ is not the Liouville current of $\phi$, but instead the Liouville current of $\mathbb H^2$ itself. And so yes, it is indeed independent of $\phi$.
Also, the object in your post labelled $\mathcal{G}(S)$ is not the space of geodesics of $S$ but is instead the space of geodesics, again, of $\mathbb H^2$ itself.
There's still some work left to do in order to apply the Liouville measure of $\mathbb H^2$ to obtain the Liouville measure associated to $\phi$. For this purpose the first thing you have to do is to choose a locally isometric universal covering map $f : \mathbb H^2 \to S$. Associated to this map you have a deck transformation action of the group $\pi_1(S)$ on $\mathbb H^2$, and that action extends continuously to an action on $\mathbb H^2 \cup \partial \mathbb H^2$. This is where the dependence on $\phi$ comes in: different choices of the hyperbolic metric $\phi$ on $S$ will produce different universal covering maps $f$, different deck transformation actions of $\pi_1(S)$ on $\mathbb H^2$, and different extensions of that action to the boundary $\partial \mathbb H^2$.
I suspect that if you go back and reread the various constructions you have seen with the concepts of the previous paragraph in mind, you will see more explicitly the dependence on $\phi$. Rather than try to guess the format of any one of those constructions, let me make a point which will be general across any of those definitions.
The key thing to focus upon is that the domain of definition of the Liouville current of $\mathbb H^2$ itself is acted upon by the deck transformation group, the quotient of that action is the domain of definition of the Liouville current of $S$ with the metric $\phi$, and the latter Liouville current is the "local pushforward" of the former with respect to that quotient.