843 research outputs found
Rigid ball-polyhedra in Euclidean 3-space
A ball-polyhedron is the intersection with non-empty interior of finitely
many (closed) unit balls in Euclidean 3-space. One can represent the boundary
of a ball-polyhedron as the union of vertices, edges, and faces defined in a
rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at
every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a
standard ball-polyhedron if its vertex-edge-face structure is a lattice (with
respect to containment). To each edge of a ball-polyhedron one can assign an
inner dihedral angle and say that the given ball-polyhedron is locally rigid
with respect to its inner dihedral angles if the vertex-edge-face structure of
the ball-polyhedron and its inner dihedral angles determine the ball-polyhedron
up to congruence locally. The main result of this paper is a Cauchy-type
rigidity theorem for ball-polyhedra stating that any simple and standard
ball-polyhedron is locally rigid with respect to its inner dihedral angles.Comment: 11 pages, 2 figure
Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces
We state that any constant curvature Riemannian metric with conical
singularities of constant sign curvature on a compact (orientable) surface
can be realized as a convex polyhedron in a Riemannian or Lorentzian)
space-form. Moreover such a polyhedron is unique, up to global isometries,
among convex polyhedra invariant under isometries acting on a totally umbilical
surface. This general statement falls apart into 10 different cases. The cases
when is the sphere are classical.Comment: Survey paper. No proof. 10 page
Variational properties of the discrete Hilbert-Einstein functional
This is a survey on rigidity and geometrization results obtained with the
help of the discrete Hilbert-Einstein functional, written for the proceedings
of the "Discrete Curvature" colloquium in Luminy.Comment: 10 page
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant
under the action of a Fuchsian group of isometries (i.e. a group of isometries
leaving globally invariant a totally geodesic surface, on which it acts
cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric
to a hyperbolic metric with conical singularities of positive singular
curvature on a compact surface of genus greater than one. We prove that these
metrics are actually realised by exactly one convex Fuchsian polyhedron (up to
global isometries). This extends a famous theorem of A.D. Alexandrov.Comment: Some little corrections from the preceding version. To appear in Les
Annales de l'Institut Fourie
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