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. TR(S0)) be the Teichmüller space (resp. reduced Teichmüller space) of S0. The length spectrum function defines a metric on TR(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

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