Results 1  10
of
22
On Teichmüller spaces of surfaces with boundary
 MR 2350850 Zbl pre05196149
"... 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 p ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
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 WeilPetersson 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 nonempty 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 counterpart of
Local Rigidity of Inversive Distance Circle Packing
 Tech. Rep. arXiv.org, Mar
"... Abstract. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle. ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Abstract. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
A variational proof of Alexandrov’s convex cap theorem; arXiv: math/0703169. [333
"... We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
(Show Context)
We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that is uniquely determined by their curvatures. 1
Hyperbolic cusps with convex polyhedral boundary
 Geom. Topol
"... We prove that a 3–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
We prove that a 3–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The proof uses the discrete total curvature functional on the space of “cusps with particles”, which are hyperbolic conemanifolds with the singular locus a union of halflines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus. Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces. 57M50; 53C24 1
On parameterizations of Teichmüller spaces of surfaces with boundary
, 2006
"... ..."
(Show Context)
Weighted triangulations for geometry processing
"... In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary t ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this paper, we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primaldual structure (through combinations of angles, areas, and lengths) and, in the process, uncover closedform expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of wellcentered meshes, selfsupporting surfaces, and sphere packing.
A NOTE ON CIRCLE PATTERNS ON SURFACES
, 2007
"... Abstract. In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). 1. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we give two different proofs of Bobenko and Springborn’s theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface up to isometry (or similarity). 1.