Generalized stretch lines for surfaces with boundary
Abstract
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodesics in the Teichmueller space of a closed/punctured surface that are not stretch lines in the sense of Thurston.
Keywords
Cite
@article{arxiv.1911.10431,
title = {Generalized stretch lines for surfaces with boundary},
author = {Daniele Alessandrini and Valentina Disarlo},
journal= {arXiv preprint arXiv:1911.10431},
year = {2021}
}
Comments
v4: Revised version, to appear on International Mathematics Research Notices. 45 pages, 24 figures v3: Exposition improved. 47 pages, 25 figures