Rigidly rotating perfect fluid stars in $2+1$ dimensions

Cataldo has found all rigidly rotating self-gravitating perfect fluid solutions in 2+1 dimensions with a negative cosmological constant $\Lambda$, for a density that is specified a priori as a function of a certain radial coordinate. We rewrite these solutions in standard polar-radial coordinates, for an arbitrary barotropic equation of state $p(\rho)$. For any given equation of state, we find the two-parameter family of solutions with a regular centre and finite total mass $M$ and angular momentum $J$ (rigidly rotating stars). For analytic equations of state, the solution is analytic except at the surface, but including at the centre. Defining the dimensionless spin $\tilde J:=\sqrt{-\Lambda}\,J$, there is precisely one solution for each $(\tilde J,M)$ in the region $|\tilde J|-1<M<|\tilde J|$, which consists of parts of the point particle region $M<-|\tilde J|$ and overspinning regions $|\tilde J|>|M|$. In an adjacent compact part of the black hole region $|\tilde J|<M$ (whose extent depends on the equation of state), there are precisely two solutions for each $(\tilde J,M)$. Hence exterior solutions exist in all three classes of BTZ solution (black hole, point particle and overspinning), but not all possible values of $(\tilde J,M)$ can be realised as stars. Regardless of the values of $\tilde J$ and $M$, the causal structure of all stars for all equations of state is that of anti-de Sitter space, without horizons or closed timelike curves.