## Abstract

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 language | English |
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Article number | 1650025 |

Number of pages | 39 |

Journal | Communications in Contemporary Mathematics |

Volume | 19 |

Issue number | 01 |

DOIs | |

Publication status | Published - 14 Jun 2016 |

MoE publication type | A1 Journal article-refereed |

## Keywords

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