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 $\{a_n\}$. And each $a_n$ 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 $\{a_n\}$ which corresponds to non arithmetic Veech surfaces is $(5-\sqrt{5})/20$, which is uniquely realized by the area of the minimal triangles of the normalized golden $L$-shaped translation surface up to affine equivalence.
Yumin ZHONG
. ON THE AREAS OF THE MINIMAL TRIANGLES IN VEECH SURFACES[J]. Acta mathematica scientia, Series B, 2020
, 40(2)
: 503
-514
.
DOI: 10.1007/s10473-020-0213-7
[1] Fox R H, Kershner R B. Geodesics on a rational polyhedron. Duke Math J, 1936, 2:147-150
[2] Gutkin E, Judge C. Affine mappings of translation surfaces:Geometry and arithmetic. Duke Math J, 2000, 103:191-213
[3] Gutkin E, Hubert P, Schmidt T. Affine diffeomorphisms of translation surfaces:periodic points, Fuchsian groups, and arithmeticity. Ann Sci École Norm Supér, 2003, 36, 847-866
[4] Hubert P, Schmidt T A. Invariants of translation surfaces. Ann Inst Fourier, 200151:461-495
[5] Kerckhoff S, Masur H, Smillie J. Ergodicity of billiard flows and quadratic differentials. Annals of Math, 1986, 124:293-311
[6] Katok A, Zemlyakov A. Topological transitivity of billiards in polygons. Math Notes, 1975, 18:760-764
[7] McMullen C T. Teichmüller curves in genus two:Discriminant and spin. Math Ann, 2005, 333:87-130
[8] McMullen C T. Billiards and Teichmüller Curves on Hilbert Modular Surfaces. J Amer Math Soc, 2003, 16(4):857-885
[9] Mukamel R. Fundamental domains and generators for lattice Veech groups. Commentarii Mathematici Helvetici, 2017, 92(1):57-83
[10] Smillie J, Weiss B. Characterizations of lattice surfaces. Invent Math, 2010, 180(3):535-557
[11] Smillie J, Weiss B. Finiteness results for flat surfaces:a survey and problem list//Partially hyperbolic dynamics, laminations, and Teichmüller flow. Toronto:Fields Institute, 2006:125-137
[12] Wu C. Lattice surfaces and smallest triangles. Geometriae Dedicata, 2017, 187(1):107-121
[13] Veech W A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent Math, 1989, 97(3):553-583
[14] Vorobets Y. Planar structures and billiards in rational polygons:the Veech alternative. Russian Math Surveys, 1996, 51:779-817
