Thurston's asymmetric metric for the space of flat metrics

Jiajun Shi

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arXiv · 2510.16852

Thurston's asymmetric metric for the space of flat metrics

Jiajun Shi

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Abstract

Thurston introduced in his seminal work an asymmetric metric on Teichm\"uller space by the ratio of simple closed curve length. In this paper, we generalize the idea and define an asymmetric metric on the space of unit-area flat metrics coming from half-translation structures on a closed surface. We also discuss two different topologies coming from the asymmetry.

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Jiajun Shi. 2025-10-19. Thurston's asymmetric metric for the space of flat metrics. https://arxiv.org/abs/2510.16852

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