Thurston's metric on Teichmüller space of semi-translation surfaces

2018/08/29 by Wolenski, Federico
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1808.09734

Abstract

The present paper is composed of two parts. In the first one we define two pseudo-metrics L_F and K_F on the Teichmuüller space of semi-translation surfaces TQ_g(\underline k,ε), which are the symmetric counterparts to the metrics defined by William Thurston on T_gⁿ. We prove some nice properties of L_F and K_F, most notably that they are complete pseudo-metrics. In the second part we define their asymmetric analogues L_F^a and K_F^a on T Q_g⁽¹⁾(k, ε) and prove that their equality depends on two statements regarding 1-Lipschitz maps between polygons. We are able to prove the first statement, but the second one remains a conjecture: nonetheless, we explain why we believe it is true.

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