Rigidity of infinite hexagonal triangulation of the plane
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- by Tianqi Wu, Xianfeng Gu and Jian Sun PDF
- Trans. Amer. Math. Soc. 367 (2015), 6539-6555 Request permission
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.References
- A. Bobenko, U. Pinkall, and B. Springborn, Discrete conformal maps and ideal hyperbolic polyhedra, arXiv:1005.2698 [math.GT], May 2010.
- Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991), no. 2, 395–412. MR 1094463
- Zheng-Xu He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1–33. MR 1680531, DOI 10.2307/121018
- Feng Luo, Private communication.
- Feng Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004), no. 5, 765–780. MR 2100762, DOI 10.1142/S0219199704001501
- Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. MR 906396
- Oded Schramm, Rigidity of infinite (circle) packings, J. Amer. Math. Soc. 4 (1991), no. 1, 127–149. MR 1076089, DOI 10.1090/S0894-0347-1991-1076089-9
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
- MR Author ID: 709542
- 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
- Received by editor(s): June 16, 2013
- Received by editor(s) in revised form: September 13, 2013
- Published electronically: November 12, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 6539-6555
- MSC (2010): Primary 52C25, 52C26
- DOI: https://doi.org/10.1090/S0002-9947-2014-06285-5
- MathSciNet review: 3356946