THE HARMONIC MAP COMPACTIFICATION OF TEICHMULLER SPACES FOR PUNCTURED RIEMANN SURFACES

In the paper [The Teichmuller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449-479], Wolf provided a global coordinate system of the Teichmuller space of a closed oriented surface S with the vector space of holomorphic quadratic differentials on a Riemann surface X homeomorphic to S. This coordinate system is via harmonic maps from the Riemann surface X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmuller space by adding a point at infinity to each endpoint of harmonic map rays starting from X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.