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On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric

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Abstract

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmüller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmüller space with the Teichmüller space of complete hyperbolic surfaces with punctures, by showing that the two spaces (endowed with the Thurston metric) are almost isometric.

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Acknowledgements

I would like to thank Lixin Liu and Weixu Su for their useful suggestions. I also thank the referee for his/her careful reading of the paper and for making numerous detailed comments and suggestions.

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  1. School of Mathematical Science, Fudan University, Shanghai, 200433, People’s Republic of China

    Huiping Pan

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  1. Huiping Pan
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Correspondence to Huiping Pan.

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This work is partially supported by NSFC, No: 11271378.

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Pan, H. On finite marked length spectral rigidity of hyperbolic cone surfaces and the Thurston metric. Geom Dedicata 191, 53–83 (2017). https://doi.org/10.1007/s10711-017-0245-x

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  • DOI: https://doi.org/10.1007/s10711-017-0245-x

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