Bull. Korean Math. Soc. 2016; 53(2): 399-410
Printed March 31, 2016
https://doi.org/10.4134/BKMS.2016.53.2.399
Copyright © The Korean Mathematical Society.
Zongliang Sun
Shenzhen University
In this paper, we study the relations between the Thurston metric and the hyperbolic metric on a closed surface of genus $g \geq 2.$ We show a rigidity result which says if there is an inequality between the marked length spectra of these two metrics, then they are isotopic. We obtain some inequalities on length comparisons between these metrics. Besides, we show certain distance distortions under conformal graftings, with respect to the Teichm\"uller metric, the length spectrum metric and Thurston's asymmetric metrics.
Keywords: complex projective structure, hyperbolic metric, marked length spectrum, Teichm\"uller space, Thurston metric
MSC numbers: Primary 30F60; Secondary 30F45, 51M25
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