It's well-known that for a Riemann sphere $S$, $Aut(S)\simeq PSL(2,\mathbb C)$. If we equip two volume forms $\omega_1,\omega_2$ on $S$, then the volume preserving isomorphism forms a subgroup of $Aut(S)$. Is there a classification about these subgroups? In particular, I am very concerned about whether there is a complex structure on these volume-preserving isomorphism groups.
For a easier case, if $\omega_1=\omega_2$, what can we say about the volume presering automorphisms?