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

TL;DR

This paper extends Wolf's harmonic maps compactification of Teichmüller spaces to punctured Riemann surfaces, demonstrating it remains equivalent to the Thurston compactification, thus broadening the understanding of these geometric structures.

Contribution

It generalizes the harmonic maps compactification to punctured surfaces and proves its equivalence with the Thurston compactification in this broader context.

Findings

Harmonic maps compactification applies to punctured Riemann surfaces.

The compactification coincides with Thurston's compactification.

The approach extends Wolf's original results to a wider class of 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.