We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle structure is defined. Both the angle structure and the volume are natural generalizations of tetrahedra in the constant sectional curvature spaces and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space. In particular, the maximum point of the

The problem is concerned about computing virtual coordinates for greedy routing in a wireless ad hoc network. Consider a set of wireless nodes S densely deployed inside a geometric domain R2. Nodes within communication range can directly communicate with each other. We ask whether one can compute a set of virtual coordinates for S such that greedy routing has guaranteed delivery. In particular, each node forwards the message to the neighbor whose distance to the destination, computed under the virtual coordinates and some metric function d, is the smallest. If such a neighbor can always be found, greedy routing successfully delivers the message to the destination. The problem can be phrased as finding a greedy embedding of S in some geometric space, such that greedy routing always succeeds. In the setting of this entry, we assume that the nodes are a dense sample of the domain such that

The purpose of this paper is to study the monodromy groups associated to the quasi-bounded holomorphic quadratic forms on punctured surfaces. As a consequence, we obtain a natural family of symplectic structures on the Teichm/iller space Tg,, for n> 0. As another consequence, we show that the projective monodromy map from a class of Fuchsian equations to the representation variety is generically a local diffeomorphism.

A spherical polyhedral surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedral surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedral surfaces in terms of the triangulation and the angle assignment is established.

We investigate the rigidity of hyperbolic cone metrics on 3manifolds which are isometric gluing of ideal and hyperideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyperideal hyperbolic polyhedral metrics. It is shown that a hyperideal hyperbolic polyhedral metric is determined up to isometry by its curvature and a decorated ideal hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. The main tool used in the proof is the Fenchel dual of the volume function.