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

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