Rigidity of infinite hexagonal triangulation of the plane

In this paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in 2004. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulatio...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2015-09, Vol.367 (9), p.6539-6555
Main Authors: Wu, Tianqi, Gu, Xianfeng, Sun, Jian
Format: Article
Language:English
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Research article
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Summary:In this paper, we consider the rigidity problem of the infinite hexagonal triangulation of the plane under the piecewise linear conformal changes introduced by Luo in 2004. Our result shows that if a geometric hexagonal triangulation of the plane is PL conformal to the regular hexagonal triangulation and all inner angles are in [δ,π/2−δ][\delta , \pi /2 -\delta ] for any constant δ>0\delta > 0, then it is the regular hexagonal triangulation. This partially solves a conjecture of Luo. The proof uses the concept of quasi-harmonic functions to unfold the properties of the mesh.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2014-06285-5