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On the Areas of the Minimal Triangles in Veech Surfaces

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Abstract

Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence {an}. And each an in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1. In this article, we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of an which corresponds to non arithmetic Veech surfaces is (5 - √5)/20, which is uniquely realized by the area of the minimal triangles of the normalized golden L-shaped translation surface up to affine equivalence.

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Acknowledgements

The author would like to thank Shengjian Wu for helpful discussions and encouragement.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Yumin Zhong

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  1. Yumin Zhong
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Correspondence to Yumin Zhong.

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Supported by National Natural Science Foundation of China (11701039) and Youth and Research and Innovation Program of BUPT (2017RC18)

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Zhong, Y. On the Areas of the Minimal Triangles in Veech Surfaces. Acta Math Sci 40, 503–514 (2020). https://doi.org/10.1007/s10473-020-0213-7

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  • DOI: https://doi.org/10.1007/s10473-020-0213-7

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