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 cone-manifolds with the singular locus a union of half-lines. 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.

3 The notion of Levi-Civita connection is recalled in Section 4. 4 Of course, if the tensor is not positive definite, it does not induce a distance on M , and in particular there is no notion of "minimization of distance" for the geodesics. 5 A smooth manifold is pseudo-Riemannian if it is endowed with a non-degenerate smooth (0, 2)-tensor. If the tensor is positive definite at each point, then the manifold is Riemannian. If the tensor has a negative direction, but no more than one linearly independent negative direction at each point, then the manifold is Lorentzian. 6 The isometry group of M is bigger than O(p, q) if M is not connected. 7 This follows from the Gauss formula and the fact that the shape operator on M is ± the identity. The fact that the sectional curvature is constant also follows from the preceding item.It is obvious that the projective quotient of M is anti-isometric to M, and will thus be denoted by M. Moreover, in the case where M is not connected, it has two connected components, which correspond under the antipodal map x → −x. In particular, M is connected.By construction, M is a subset of the projective space RP n . Indeed, topologically M can be defined asor the same definition with > replaced by <, depending on the case. Hence any choice of an affine hyperplane in R n+1 which does not contain the origin will give an affine chart of the projective space, and the image of M in the affine chart will be an open subset of an affine space of dimension n.Recall that PGL(n + 1, R), the group of projective transformations (or homographies), is the quotient of GL(n + 1, R) by the non-zero scalar transformations. Even if M is not connected, an isometry of M passes to the quotient only if it is an element of O(p, q). Hence Isom(M), the isometry group of M, is PO(p, q), the quotient of O(p, q) by {±Id} (indeed elements of O(p, q) have determinant equal to 1 or −1). Lines and pseudo-distancePseudo-distance on M. A geodesic c of M is the non-empty intersection of M with a linear plane. This intersection might not be connected. The geodesic c of M is said to be

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.

The hyperbolic space H d can be defined as a pseudo-sphere in the pd`1q Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that H d {Γ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn-Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere.The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov-Fenchel and Brunn-Minkowski inequalities. Here the inequalities are reversed. IntroductionThere are two main motivations behind the definitions and results presented here. See next section for a precise definition of Fuchsian convex bodies, the main object of this paper, and Fuchsian convex surfaces (boundaries of Fuchsian convex bodies).The first motivation is to show that the geometry of Fuchsian convex surfaces in the Minkowski space is the right analogue of the classical geometry of convex compact hypersurfaces in the Euclidean space. In the present paper, we show the analogue of the basics results of what is called Brunn-Minkowski theory. Roughly speaking, the matter is to study the relations between the sum and the volume of the bodies under consideration. Actually here we associate to each convex set the volume of another region of the space, determined by the convex set, so we will call it the covolume of the convex set. This generalization is as natural as, for example, going from the round sphere to compact hyperbolic surfaces. To strengthen this idea, existing results can be put into perspective. Indeed, Fuchsian convex surfaces are not new objects. As far I know, smooth Fuchsian hypersurfaces appeared in [Oliker and Simon, 1983], see Subsection 3.3. The simplest examples of convex Fuchsian surfaces are convex hulls of the orbit of one point for the action of the Fuchsian group. They were considered in [Näätänen and Penner, 1991], in relation with the seminal papers [Penner, 1987, Epstein andPenner, 1988]. See also [Charney et al., 1997]. The idea is to study hyperbolic problems via the extrinsic structure given by the Minkowski space. For a recent illustration see [Espinar et al., 2009]. The first study of Fuchsian surfaces for their own is probably [Labourie and Schlenker, 2000]. The authors proved that for any Riemannian metric on a compact surface of genus ě 2 with negative curvature, there exists an isometric convex Fuchsian surface in the 2`1-Minkowski space, up to a quotient. In the Euclidean case, the analog problem is known as Weyl problem. A uniqueness result is also given. This kind of result about realization of abstract metrics by ...