## Geometry & Topology

### Rigidity of polyhedral surfaces, II

#### Abstract

We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

#### Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1265-1312.

Dates
Accepted: 17 January 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800247

Digital Object Identifier
doi:10.2140/gt.2009.13.1265

Mathematical Reviews number (MathSciNet)
MR2496046

Zentralblatt MATH identifier
1160.52012

#### Citation

Guo, Ren; Luo, Feng. Rigidity of polyhedral surfaces, II. Geom. Topol. 13 (2009), no. 3, 1265--1312. doi:10.2140/gt.2009.13.1265. https://projecteuclid.org/euclid.gt/1513800247

#### References

• A I Bobenko, B A Springborn, Variational principles for circle patterns and Koebe's theorem, Trans. Amer. Math. Soc. 356 (2004) 659–689
• W Brägger, Kreispackungen und Triangulierungen, Enseign. Math. $(2)$ 38 (1992) 201–217
• B Chow, F Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003) 97–129
• Y Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991) 655–669
• W Fenchel, J Nielsen, Discontinuous groups of isometries in the hyperbolic plane, de Gruyter Studies in Math. 29, Walter de Gruyter & Co., Berlin (2003) Edited and with a preface by A L Schmidt, Biography of the authors by B Fuglede
• R Guo, On parameterizations of Teichmüller spaces of surfaces with boundary, to appear in J. Differential Geom.
• G P Hazel, Triangulating Teichmüller space using the Ricci flow, PhD thesis, University of California San Diego (2004) Available at \setbox0\makeatletter\@url www.math.ucsd.edu/~thesis/thesis/ghazel/ghazel.pdf \unhbox0
• G Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002) 361–391
• F Luo, Rigidity of polyhedral surfaces
• F Luo, A characterization of spherical polyhedral surfaces, J. Differential Geom. 74 (2006) 407–424
• F Luo, On Teichmüller spaces of surfaces with boundary, Duke Math. J. 139 (2007) 463–482
• G Mondello, Triangulated Riemann surfaces with boundary and the Weil–Petersson Poisson structure, J. Differential Geom. 81 (2009) 391–436
• R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339
• I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. $(2)$ 139 (1994) 553–580
• J-M Schlenker, Circle patterns on singular surfaces, Discrete Comput. Geom. 40 (2008) 47–102
• K Stephenson, Introduction to circle packing: The theory of discrete analytic functions, Cambridge Univ. Press (2005)
• W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
• W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Series 35, Princeton University Press (1997) Edited by S Levy