Minkowski-type and Alexandrov-type theorems for polyhedral herissons

Smoothing 3-dimensional polyhedral spaces

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.

致 Alexandrov 几何：CAT(0) 空间

An invitation to Alexandrov geometry: CAT(0) spaces

本文通过最少的准备，展示 Alexandrov 几何的美丽与强大，涵盖 CAT(0) 空间的多个应用和定理。

Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes

We present a constructive proof of Alexandrov's theorem regarding the existence of a convex polytope with a given metric on the boundary. The polytope is obtained as a result of a certain deformation in the class of generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a relation with the weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of a positively curved generalized convex polytope. The latter is shown to be e…

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter co…

具有多面体类型边界的双曲3-流形

Hyperbolic 3-manifolds with boundary of polyhedral type

本文研究具有双曲内部和凸边界的紧致3-流形，证明边界上凸双曲锥度量可由弯曲度量实现且可控多面体实现唯一，并得出凸余紧度量的全局与无穷小刚性，推进Thurston猜想。

Reshaping Convex Polyhedra

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset…