A rigidity criterion for non-convex polyhedra

Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra, which deals with deformations that preserve face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volum…

On the infinitesimal rigidity of polyhedra with vertices in convex position

Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In partic…

多面体的初始数据集刚性结果

Initial data set rigidity results for polyhedra

本文利用旋量证明了多面体初始数据集的二面角型刚性，将时空正质量定理的刚性分析扩展至扭曲旋量丛设定，揭示了时空正质量定理、二面角刚性与带毛细管的边缘被困曲面之间的联系。

On the infinitesimal rigidity of weakly convex polyhedra

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by ``denting'' at most two edges at a common vertex, and suspensions with a natural subdivision.

无限圆图案和双曲三维空间中凸多面体的特征化

Characterizations of infinite circle patterns and convex polyhedra in hyperbolic 3-space

本文研究了双曲三维空间中无限圆图案和凸多面体的特征化问题，证明了当0≤Θ<π时无限圆图案的刚性和统一化定理，并解决了正则圆图案的分类问题。

Deformations of hyperbolic convex polyhedra and 3-cone-manifolds

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we prove that local rigidity holds and obtain that the space of convex hyperbolic polyhedra with given combinatorial type is locally par…