I'm starting to study geodesic currents and I have a question concerning uniform continuity.

Let's take $S$ a closed surface of genus $g$ and $C(S)$ the space of geodesic currents on $S$ (as it is defined by Bonahon, endowed with the weak star topology). Let's call $\mathcal{C}(S)$ the set of isotopy classes of closed curves on $S$ and $\mathcal{S}\subset\mathcal{C}(S)$ the subset corresponding to simple closed curves.

It is known that $\mathcal{C}(S)$ injects in $C(S)$ and that there is a continuous extension $i:C(S)\times C(S)\rightarrow \mathbb{R}$ of the intersection number on $\mathcal{C}(S)$ .

Writing $T(S)$ for the Teichmuller space of $S$, it is also known that there is an injection $T(S)\rightarrow C(S)$ which sends a hyperbolic metric $h$ to the Liouville current $L_h$. For every Liouville current $L_h$ and every $\alpha\in \mathcal{S}$ it is true $l_h(\alpha)=i(L_h,\alpha)$.

My question is: I know that, for every $h\in T(S)$ hyperbolic metric, that the function $i(L_h,-):\mathcal{S}\rightarrow \mathbb{R}^+$ is continuous, but is it also $uniformly$ continuous?