The harmonic maps compactification of Teichm\"uller spaces for punctured Riemann surfaces

Abstract

Wolf gave a homeomorphism from the Teichm\"uller space to the space of quadratic differentials on a closed Riemann surface by using harmonic maps. Moreover, using harmonic maps rays, he gave a compactification of the Teichm\"uller space and show that it coincides with the Thurston compactification. In this paper, we extend the harmonic maps compactification to the Teichm\"uller spaces of punctured Riemann surfaces, and show that it still coincides with the Thurston compactification.