## On Teichmüller spaces of surfaces with boundary

Venue: | MR 2350850 Zbl pre05196149 |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Luo_onteichmüller,

author = {Feng Luo},

title = {On Teichmüller spaces of surfaces with boundary},

booktitle = {MR 2350850 Zbl pre05196149},

year = {},

pages = {463--482}

}

### OpenURL

### Abstract

We characterize hyperbolic metrics on compact triangulated surfaces with boundary using a variational principle. As a consequence, a new parameterization of the Teichmüller space of compact surface with boundary is produced. In the new parameterization, the Teichmüller space becomes an open convex polytope. It is conjectured that the Weil-Petersson symplectic form can be expressed explicitly in terms of the new coordinate. 1.1. The purpose of this paper is to produce a new parameterization of the Teichmüller space of compact surface with non-empty boundary so that the lengths of the boundary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counter-part of

### Citations

357 |
Intersection theory on the moduli space of curves and the matrix airy function
- Kontsevich
- 1992
(Show Context)
Citation Context ...ringborn [BS] using discrete integrable systems. Parameterization of Teichmüller space using metric ribbon graph has been used extensively recently. See for instance the solution of Witten conjecture =-=[Ko]-=- and the stability of the homology of the mapping class group [Ha]. In the metric ribbon graph approach, the key lies in the singular flat metrics arising from Jenkins-Strebel differentials. The appro... |

180 |
Geometry and Spectra of compact Riemann Surfaces
- Buser
- 1992
(Show Context)
Citation Context ...tion of the Weil-Petersson volume of the moduli spaces of curves in [Mi]. Using a 3-holed sphere decomposition of the surface S and the associated Fenchel-Nielsen coordinate, it is known (see [IT] or =-=[Bu]-=-) that T(S,l) is diffeomorphic to (R × R>0) N for some integer N. One can decompose the surface S into a union of hexagons instead of 3-holed spheres. These decompositions are called ideal triangulati... |

127 |
Convex Analysis and Nonlinear Optimization
- Borwein, Lewis
- 2006
(Show Context)
Citation Context ... ∑n i=1 yizi| y ∈ D }. The dual linear programming problem is max{0|x ∈ L(S, T, z)} by the construction. By the duality theorem of linear programming (in fact Fakars lemma suffices in this case) (see =-=[BL]-=- for instance), L(S, T, z) ̸= ∅ if and only if for each non-zero vector y ∈ D, ∑n i=1 yizi > 0. The set D is a cone in Rn . Furthermore, the inequalities yi + yj ≥ yk, yk + yi ≥ yj and yj + yk ≥ yi im... |

118 |
The Geometry and Topology of 3-manifolds, lecture notes
- Thurston
(Show Context)
Citation Context ...aper. Theorem 1.2 can be deduced from theorem 1.1 by analyzing the map sending the length coordinate to the E-coordinate. It is motivated by Thurston’s original proof of the circle packing theorem in =-=[Th]-=-. 1.6. The techniques used in the paper are related to and motivated by the beautiful variational principles developed by Colin de Verdiere [CV1], Brägger [Br], Rivin [Ri] and Leibon [Le] for circle p... |

115 |
Stability of the homology of the mapping class groups of orientable surfaces
- Harer
- 1985
(Show Context)
Citation Context ...of Teichmüller space using metric ribbon graph has been used extensively recently. See for instance the solution of Witten conjecture [Ko] and the stability of the homology of the mapping class group =-=[Ha]-=-. In the metric ribbon graph approach, the key lies in the singular flat metrics arising from Jenkins-Strebel differentials. The approach in this paper can be considered as a counter-part of metric ri... |

106 |
The decorated Teichmüller space of punctured surfaces
- Penner
- 1987
(Show Context)
Citation Context ...per can be considered as a counter-part of metric ribbon graph theory using hyperbolic metrics instead of flat metrics. Another approach using hyperbolic geometry has been worked out by Penner [Pe1], =-=[Pe2]-=-. There are many works on constructing coordinates for Teichmüller spaces. In the 4work of Bonahon [Bo], a parameterization of the Teichmüller space of compact surface with boundary was produced usin... |

68 | Euclidean structures on simplicial surfaces and hyperbolic volume - Rivin - 1994 |

56 | Variational principles for circle patterns and Koebe’s theorem
- Bobenko, Springborn
(Show Context)
Citation Context ...nce, quantum Teichmüller theory) is not clear to us. Another rich source of energy functions for variational principles on triangulated surfaces has been discovered recently by Bobenko and Springborn =-=[BS]-=- using discrete integrable systems. Parameterization of Teichmüller space using metric ribbon graph has been used extensively recently. See for instance the solution of Witten conjecture [Ko] and the ... |

35 |
Weil–Petersson volumes and intersection theory on the moduli space of curves, preprint
- Mirzakhani
- 2003
(Show Context)
Citation Context ...f those isotopy classes of metrics so that the length of bi in the metrics is li. The space T(S,l) has been used recently in calculation of the Weil-Petersson volume of the moduli spaces of curves in =-=[Mi]-=-. Using a 3-holed sphere decomposition of the surface S and the associated Fenchel-Nielsen coordinate, it is known (see [IT] or [Bu]) that T(S,l) is diffeomorphic to (R × R>0) N for some integer N. On... |

30 | Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic
- Bonahon
- 1996
(Show Context)
Citation Context ...omponent (1.2) k∑ z(ei) = lj. i=1 It seems highly likely that the Weil-Petersson symplectic form on the bordered Teichmüller space T(S,l) can be expressed explicitly in terms of the E-coordinate. See =-=[Bo]-=-. 1.5. The strategy of proving theorems 1.1 and 1.2 goes as follows. By a length structure on the ideal triangulated surface (S,T) we mean a map x : C(S,T) → R>0 assigning each x-arc a positive number... |

21 | Decorated Teichmüller theory of bordered surfaces. Preprint math.GT/0210326. MathReview [SW
- Penner
(Show Context)
Citation Context ...this paper can be considered as a counter-part of metric ribbon graph theory using hyperbolic metrics instead of flat metrics. Another approach using hyperbolic geometry has been worked out by Penner =-=[Pe1]-=-, [Pe2]. There are many works on constructing coordinates for Teichmüller spaces. In the 4work of Bonahon [Bo], a parameterization of the Teichmüller space of compact surface with boundary was produc... |

20 | Characterizing the Delaunay decompositions of compact hyperbolic surfaces. math.GT/0103174
- Leibon
- 2001
(Show Context)
Citation Context ...ths of the boundary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counter-part of the work of =-=[Le]-=-, [Ri] and [Lu1] for hyperbolic, Euclidean and spherical cone metrics on closed surfaces. In these approaches, constant curvature metrics are identified with the critical points of some natural energy... |

15 |
Tiling the projective foliation space of a punctured surface
- Mosher
- 1988
(Show Context)
Citation Context ...the edges of a 2-cell}. This shows that D can be identified with the space of all measured laminations on the surface S where yi’s are the geometric intersection coordinates. By the work of [Th] (see =-=[Mo]-=- or others), it is known that every measured lamination on S considered as a vector in D is a non-negative linear combination of those vectors in D associated to essential simple loops. Furthermore, t... |

14 |
Kreispackungen und Triangulierungen
- Brägger
- 1992
(Show Context)
Citation Context ...l proof of the circle packing theorem in [Th]. 1.6. The techniques used in the paper are related to and motivated by the beautiful variational principles developed by Colin de Verdiere [CV1], Brägger =-=[Br]-=-, Rivin [Ri] and Leibon [Le] for circle packing, singular Euclidean and singular hyperbolic structures on surfaces. In these works, the energy functions are all related to the 3-dimensional volume. (T... |

13 | Rigidity of polyhedral surfaces
- Luo
(Show Context)
Citation Context ...and the maximum point of V | : L(S,T,z) → R always exists. The necessity of conditions (1.1) and (1.2) can be verified easily. We remark that there are now different proofs of theorems 1.1 and 1.2 in =-=[Lu3]-=-. The new proof of theorem 1.1 uses the Legendre transform of the energy function used in this paper. Theorem 1.2 can be deduced from theorem 1.1 by analyzing the map sending the length coordinate to ... |

8 |
Un principe variationnel pour les empilements de cercles
- Verdiere, Yves
- 1991
(Show Context)
Citation Context ...,T) we mean a map x : C(S,T) → R>0 assigning each x-arc a positive number. Length structure is the counter-part of angle structure on closed triangulated surfaces first introduced by Coin de Verdiere =-=[CV1]-=- and Rivin 3[Ri]. The E-invariant of a length structure x is the function Dx : E → R assigning each edge e the value 1 2 (∑w∈I x(w) −∑w ′ ∈II x(w′ )) where I is the set of all x-arcs adjacent to e an... |

8 | Volume and angle structures on 3-manifolds
- Luo
(Show Context)
Citation Context ...k = sinh(yi)sinhxj sinhxk, (b) ∂yi ∂xi (c) ∂yi ∂xj = sinh(xi) Aijk = − ∂yi ∂xi = Asinh(yi) where A > 0 is independent of indices, cosh yk. The proof is a simple exercise in calculus, see for instance =-=[Lu2]-=-. 5Introduce a new variable ti = (xj + xk − xi)/2 for {i,j,k} = {1,2,3}. Then xi = tj + tk. The space of all colored hyperbolic right-angled hexagons parameterized by the new coordinate t = (t1,t2,t3... |

6 | A variational principle for weighted Delaunay triangulations and hyperideal polyhedra
- Springborn
(Show Context)
Citation Context ...e of ideally triangulated surfaces with boundary. The λ-coordinate and the coordinate introduced in this paper are quite different. Other related works are the papers of M. Schlenker [Sc], Springborn =-=[Sp]-=- and Ushijima [Us]. The relationship between the result in this paper and works of Bonahon, Penner and Schlenker is not clear to us. This deserves a further study. A fascinating question, suggested by... |

5 |
An introduction to Teichmüller spaces, Translated and revised from the Japanese by the authors
- Imayoshi, Y, et al.
- 1992
(Show Context)
Citation Context ... calculation of the Weil-Petersson volume of the moduli spaces of curves in [Mi]. Using a 3-holed sphere decomposition of the surface S and the associated Fenchel-Nielsen coordinate, it is known (see =-=[IT]-=- or [Bu]) that T(S,l) is diffeomorphic to (R × R>0) N for some integer N. One can decompose the surface S into a union of hexagons instead of 3-holed spheres. These decompositions are called ideal tri... |

5 |
A canonical cellular decomposition of the Teichmüller space of compact surfaces with boundary
- Ushijima
- 1999
(Show Context)
Citation Context ...gulated surfaces with boundary. The λ-coordinate and the coordinate introduced in this paper are quite different. Other related works are the papers of M. Schlenker [Sc], Springborn [Sp] and Ushijima =-=[Us]-=-. The relationship between the result in this paper and works of Bonahon, Penner and Schlenker is not clear to us. This deserves a further study. A fascinating question, suggested by a referee, is whe... |

4 | Hyperideal polyhedra in hyperbolic manifolds
- Schlenker
- 2002
(Show Context)
Citation Context ... Teichmüller space of ideally triangulated surfaces with boundary. The λ-coordinate and the coordinate introduced in this paper are quite different. Other related works are the papers of M. Schlenker =-=[Sc]-=-, Springborn [Sp] and Ushijima [Us]. The relationship between the result in this paper and works of Bonahon, Penner and Schlenker is not clear to us. This deserves a further study. A fascinating quest... |

2 |
private communication
- Verdiere, Yves
(Show Context)
Citation Context ...rfaces. In these works, the energy functions are all related to the 3-dimensional volume. (The energy functional used by Colin de Verdiere was discovered by using the Schlaefli formula for tetrahedra =-=[CV2]-=-.) In [Lu3], we observe that all these energy functions can be constructed using the cosine law and Legendre transform. Furthermore, the cosine law produces continuous families of energy functions for... |

1 |
A Characterization of spherical polyhedral surfaces
- Luo
(Show Context)
Citation Context ...dary components are fixed. In this new parameterization, the Teichmüller space becomes an explicit open convex polytope. Our result can be considered as the counter-part of the work of [Le], [Ri] and =-=[Lu1]-=- for hyperbolic, Euclidean and spherical cone metrics on closed surfaces. In these approaches, constant curvature metrics are identified with the critical points of some natural energy functions. The ... |

1 |
A Characterization of
- Luo
(Show Context)
Citation Context ...undary components are fixed. In this new parametrization, the Teichmuller space becomes an explict open convex polytope. Our result can be considered as the counter-part of the work of [Le], [Ri] and =-=[Lu1]-=- for hyperbolic, Euclidean and spherical cone metrics on closed surfaces. In these approaches, constant curvature metrics are identified with the critical points of some natural energy functions. The ... |

1 |
A canonical cellular decomposition of the Teichmller space of compact surfaces with boundary
- Ushijima
- 1999
(Show Context)
Citation Context ...ng shearing cocycles. It is not clear to us the the exact relationship between our result and the Bonahon-Thurston shearing coordinates of the Teichmuller space. See also the related work of [Pe] and =-=[Us]-=-. As a convention in this paper, all surfaces are assumed to be connected compact with non-empty boundary and have negative Euler characteristic unless mentioned otherwise. A hyperbolic metric on the ... |