论文标题
多面体表面上顶点缩放的全球刚度的新证明
A new proof for global rigidity of vertex scaling on polyhedral surfaces
论文作者
论文摘要
Luo引入了多面体表面上的分段线性指标的顶点缩放，后者通过建立变异原理并猜想了全球刚度，证明了局部刚度。 Luo的猜想是由Bobenko-Pinkall-Springborn解决的，Bobenko-Pinkall-Springborn也引入了分段双曲指标的顶点缩放，并证明了其全球刚度。 Bobenko-Pinkall-Spingborn的证明是基于他们观察到顶点缩放和Polyhedra在$ 3 $尺寸的双曲线空间以及理想和高理想四面体的凹陷中的几何形状的观察。在本文中，我们给出了基础和简短的变分证据，证明了顶点缩放的全局刚度，而无需涉及$ 3 $维的双曲线几何形状。该方法基于矩阵特征值的连续性和凸函数的扩展。
The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo, who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo's conjecture was solved by Bobenko-Pinkall-Springborn, who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn's proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in $3$-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. In this paper, we give an elementary and short variational proof of the global rigidity of vertex scaling without involving $3$-dimensional hyperbolic geometry. The method is based on continuity of eigenvalues of matrices and the extension of convex functions.
