An analytic construction of the Deligne-Mumford compactification of the moduli space of curves

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves Mg,n. Their compactification M̄g,n is a projective algebraic variety, and as such it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmüller space by the action of the mapping class group gives a compactification of Mg,n. We put an analytic structure on this quotient and prove that with respect to this structure, the compactification is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification M̄g,n.