Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 39, 2014, 513-526

MODIFIED LENGTH SPECTRUM METRIC ON THE TEICHMÜLLER SPACE OF A RIEMANN SURFACE WITH BOUNDARY

Jun Hu and Francisco G. Jimenez-Lopez

Brooklyn College of CUNY, Department of Mathematics
Brooklyn, NY 11210, U.S.A.; junhu 'at' brooklyn.cuny.edu
and Graduate Center of CUNY, Ph.D. Program in Mathematics
365 Fifth Avenue, New York, NY 10016, U.S.A.; JHu1 'at' gc.cuny.edu

CUNY Graduate Center, Department of Mathematics
365 Fifth Avenue, New York, NY 10016, U.S.A.; fjimenez_lopez 'at' gc.cuny.edu

Abstract. Let S₀ be a bordered Riemann surface of finite type, and let T(S₀) (resp. T^R(S₀)) be the Teichmüller space (resp. reduced Teichmüller space) of S₀. The length spectrum function defines a metric on T^R(S₀)) but not on T(S₀). In this paper, we introduce a modified length spectrum function that does define a metric on T(S₀). Then we show that if two points of T(S₀) are close in the Teichmüller metric then they are close in the modified length spectrum metric, but the converse is not true. We also prove that T(S₀) is not complete under this modified length spectrum metric.

2010 Mathematics Subject Classification: Primary 30F60.

Key words: Teichmüller space, Length spectrum, Teichmüller metric.

Reference to this article: J. Hu and F.G. Jimenez-Lopez: Modified length spectrum metric on the Teichmüller space of a Riemann surface with boundary. Ann. Acad. Sci. Fenn. Math. 39 (2014), 513-526.

Full document as PDF file

doi:10.5186/aasfm.2014.3945