Interpreting Mass via Riemannian Polyhedra

Mass and Riemannian Polyhedra

We show that the concept of the ADM mass in general relativity can be understood as the limit of the total mean curvature plus the total defect of dihedral angle of the boundary of large Riemannian polyhedra. We also express the $n$-dimensional mass as a suitable integral of geometric quantities that determine the $(n-1)$-dimensional mass.

用多面体逼近黎曼流形

Approximating Riemannian manifolds by polyhedra

本文在受限黎曼框架下研究用多面体逼近黎曼流形，并沿 Regge 的原始直观思路给出 Regge 定理的证明，说明标量曲率积分可由多面体曲率逼近。

An existence result of energy minimizer maps between riemannian polyhedra

In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of riemannian polhyedra without focal points. Among other things, we show that on the relevant polyhedra, there exists a convex supporting function.

Hölder continuity of energy minimizer maps between riemannian polyhedra

The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete our existence theorem obtained in [5], and consequently we generalize completely, to the case of target polyhedra without focal points (which is weaker geometric condition than the nonpositivity of the curvature), the Eells-fuglede's existence and regularity theorem [12, chapters 1…

Polyhedral approximations of Riemannian manifolds

We give a condition on the curvature tensors of Riemannian manifolds that admit Lipschitz approximation by polyhedral metrics with curvature bounded below or above. We show that this condition is also sufficient for the existence of local approximations. We conjecture that it is also sufficient for the global approximations and prove it in some special cases.

关于正数量曲率3-流形中具有非钝二面角的黎曼多面体

On Riemannian polyhedra with non-obtuse dihedral angles in 3-manifolds with positive scalar curvature

本文确定了R^3中三维简单凸多面体的组合类型，其能在正数量曲率黎曼3-流形中实现为具非钝二面角的黎曼多面体，并视为Andreev定理的类比，同时构造了高维例子。