43 research outputs found
Polyhedral hyperbolic metrics on surfaces
In the last section of \cite{CompHyp} it is proved that the map
is a finite-sheeted covering map between and . As
is simply connected it is deduced that is a
homeomorphism. The fact that is connected is missing. Here we
provide a proof
Polygons of the Lorentzian plane and spherical simplexes
It is known that the space of convex polygons in the Euclidean plane with
fixed normals, up to homotheties and translations, endowed with the area form,
is isometric to a hyperbolic polyhedron. In this note we show a class of convex
polygons in the Lorentzian plane such that their moduli space, if the normals
are fixed and endowed with a suitable area, is isometric to a spherical
polyhedron. These polygons have an infinite number of vertices, are space-like,
contained in the future cone of the origin, and setwise invariant under the
action of a linear isometry.Comment: New text, title slightly change
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
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
Fuchsian polyhedra in Lorentzian space-forms
Let S be a compact surface of genus >1, and g be a metric on S of constant
curvature K\in\{-1,0,1\} with conical singularities of negative singular
curvature. When K=1 we add the condition that the lengths of the contractible
geodesics are >2\pi. We prove that there exists a convex polyhedral surface P
in the Lorentzian space-form of curvature K and a group G of isometries of this
space such that the induced metric on the quotient P/G is isometric to (S,g).
Moreover, the pair (P,G) is unique (up to global isometries) among a particular
class of convex polyhedra, namely Fuchsian polyhedra. This extends theorems of
A.D. Alexandrov and Rivin--Hodgson concerning the sphere to the higher genus
cases, and it is also the polyhedral version of a theorem of
Labourie--Schlenker
Gauss images of hyperbolic cusps with convex polyhedral boundary
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral
boundary is uniquely determined by its Gauss image. Furthermore, any spherical
metric on the torus with cone singularities of negative curvature and all
closed contractible geodesics of length greater than is the metric of
the Gauss image of some convex polyhedral cusp. This result is an analog of the
Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in
terms of their Gauss images.
The proof uses a variational method. Namely, a cusp with a given Gauss image
is identified with a critical point of a functional on the space of cusps with
cone-type singularities along a family of half-lines. The functional is shown
to be concave and to attain maximum at an interior point of its domain. As a
byproduct, we prove rigidity statements with respect to the Gauss image for
cusps with or without cone-type singularities.
In a special case, our theorem is equivalent to existence of a circle pattern
on the torus, with prescribed combinatorics and intersection angles. This is
the genus one case of a theorem by Thurston. In fact, our theorem extends
Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's
theorem on compact convex polyhedra with non-obtuse dihedral angles.
The functional used in the proof is the sum of a volume term and curvature
term. We show that, in the situation of Thurston's theorem, it is the potential
for the combinatorial Ricci flow considered by Chow and Luo.
Our theorem represents the last special case of a general statement about
isometric immersions of compact surfaces.Comment: 55 pages, 17 figure
The equivariant Minkowski problem in Minkowski space
The classical Minkowski problem in Minkowski space asks, for a positive
function on , for a convex set in Minkowski space with
space-like boundary , such that is the
Gauss--Kronecker curvature at the point with normal . Analogously to the
Euclidean case, it is possible to formulate a weak version of this problem:
given a Radon measure on the generalized Minkowski problem
in Minkowski space asks for a convex subset such that the area measure of
is .
In the present paper we look at an equivariant version of the problem: given
a uniform lattice of isometries of , given a
invariant Radon measure , given a isometry group of
Minkowski space, with as linear part, there exists a unique convex set
with area measure , invariant under the action of .
The proof uses a functional which is the covolume associated to every
invariant convex set.
This result translates as a solution of the Minkowski problem in flat space
times with compact hyperbolic Cauchy surface. The uniqueness part, as well as
regularity results, follow from properties of the Monge--Amp\`ere equation. The
existence part can be translated as an existence result for Monge--Amp\`ere
equation.
The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for
and by V.~Oliker and U.~Simon for . Our method is
totally different. Moreover, we show that those cases are very specific: in
general, there is no smooth -invariant surface of constant
Gauss-Kronecker curvature equal to
Lorentzian area measures and the Christoffel problem
We introduce a particular class of unbounded closed convex sets of
, called F-convex sets (F stands for future). To define them, we use
the Minkowski bilinear form of signature instead of the usual
scalar product, and we ask the Gauss map to be a surjection onto the hyperbolic
space \H^d. Important examples are embeddings of the universal cover of
so-called globally hyperbolic maximal flat Lorentzian manifolds.
Basic tools are first derived, similarly to the classical study of convex
bodies. For example, F-convex sets are determined by their support function,
which is defined on \H^d. Then the area measures of order , are defined. As in the convex bodies case, they are the coefficients of the
polynomial in which is the volume of an approximation of
the convex set. Here the area measures are defined with respect to the
Lorentzian structure.
Then we focus on the area measure of order one. Finding necessary and
sufficient conditions for a measure (here on \H^d) to be the first area
measure of a F-convex set is the Christoffel Problem. We derive many results
about this problem. If we restrict to "Fuchsian" F-convex set (those who are
invariant under linear isometries acting cocompactly on \H^d), then the
problem is totally solved, analogously to the case of convex bodies. In this
case the measure can be given on a compact hyperbolic manifold.
Particular attention is given on the smooth and polyhedral cases. In those
cases, the Christoffel problem is equivalent to prescribing the mean radius of
curvature and the edge lengths respectively