Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 349-360

ON THE LENGTH SPECTRUM METRIC IN INFINITE DIMENSIONAL TEICHMÜLLER SPACES

Erina Kinjo

Tokyo Institute of Technology, Department of Mathematical and Computing Sciences
Oh-okayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan; kinjo.e.aa 'at' m.titech.ac.jp

Abstract. We consider the length spectrum metric d_L in infinite dimensional Teichmüller space T(R₀). It is known that d_L defines the same topology as that of the Teichmüller metric d_T on T(R₀) if R₀ is a topologically finite Riemann surface. In 2003, Shiga proved that d_L and d_T define the same topology on T(R₀) if R₀ is a topologically infinite Riemann surface which can be decomposed into pairs of pants such that the lengths of all their boundary components except punctures are uniformly bounded by some positive constants from above and below. In this paper, we extend Shiga's result to Teichmüller spaces of Riemann surfaces satisfying a certain geometric condition.

2010 Mathematics Subject Classification: Primary 30F60; Secondary 32G15.

Key words: Length spectrum, Teichmüller metric, Riemann surface of infinite type.

Reference to this article: E. Kinjo: On the length spectrum metric in infinite dimensional Teichmüller spaces. Ann. Acad. Sci. Fenn. Math. 39 (2014), 349-360.

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doi:10.5186/aasfm.2014.3925