Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}(S)$ of $S$. These metrics induce respective volume forms $vol(g)$ and $vol(g')$ on the unit tangent bundle $UTS$.
Pardon my ignorance, but what can we say on whether or not these two volume forms are absolutely continuous to each other?
Is there a way to answer this question using standard techniques/results from Teichmuller theory or dynamics?
