Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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.

Some remarks on Thurston metric and hyperbolic metric

Zongliang Sun

Shenzhen University

Abstract

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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