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Rigidity of infinite hexagonal triangulation of the plane


Authors: Tianqi Wu, Xianfeng Gu and Jian Sun
Journal: Trans. Amer. Math. Soc. 367 (2015), 6539-6555
MSC (2010): Primary 52C25, 52C26
Published electronically: November 12, 2014
MathSciNet review: 3356946
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Abstract: 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 $ [\delta , \pi /2 -\delta ]$ for any constant $ \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.


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Additional Information

Tianqi Wu
Affiliation: Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email: mike890505@gmail.com

Xianfeng Gu
Affiliation: Department of Computer Science, Stony Brook University, Stony Brook, New York 11794
Email: gu@cs.stonybrook.edu

Jian Sun
Affiliation: Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China
Email: jsun@math.tsinghua.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2014-06285-5
Received by editor(s): June 16, 2013
Received by editor(s) in revised form: September 13, 2013
Published electronically: November 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society