All static circularly symmetric perfect fluid solutions of $2+1$ gravity

Via a straightforward integration of the Einstein equations with a cosmological constant, all static circularly symmetric perfect fluid $2+1$ solutions are derived. The structural functions of the metric depend on the energy density, which remains, in general, arbitrary. Spacetimes for fluids satisfying linear and polytropic state equations are explicitly derived; they describe, among others, stiff matter, monatomic and diatomic ideal gases, nonrelativistic degenerate fermions, incoherent and pure radiation. As a by-product, we demonstrate the uniqueness of the constant energy density perfect fluid within the studied class of metrics. A full similarity of the perfect fluid solutions with a constant energy density of the $2+1$ and $3+1$ gravities is established.