Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

David Radnell*, Eric Schippers, Wolfgang Staubach

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

12 Citations (Scopus)
88 Downloads (Pure)


Consider a Riemann surface of genus (Formula presented.) bordered by (Formula presented.) curves homeomorphic to the unit circle, and assume that (Formula presented.). For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in (Formula presented.) and (Formula presented.), and furthermore that the space of (Formula presented.) differentials in (Formula presented.) decomposes as a direct sum of infinitesimally trivial differentials and (Formula presented.) harmonic (Formula presented.) differentials. Thus the tangent space of this Teichmüller space is given by (Formula presented.) harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of (Formula presented.) harmonic Beltrami differentials.

Original languageEnglish
Article number1650025
Number of pages39
JournalCommunications in Contemporary Mathematics
Issue number01
Publication statusPublished - 14 Jun 2016
MoE publication typeA1 Journal article-refereed


  • (Formula presented.) Beltrami differentials
  • bordered Riemann surfaces
  • Gardiner–Schiffer variation
  • infinitesimally trivial Beltrami differentials
  • Teichmüller theory
  • Weil–Petersson metric


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