A new proof for global rigidity of vertex scaling on polyhedral surfaces

TL;DR

This paper presents a simplified, elementary proof of the global rigidity of vertex scaling on polyhedral surfaces, avoiding complex hyperbolic geometry by using matrix eigenvalue continuity and convex function extension.

Contribution

It provides a shorter, more accessible proof of global rigidity for vertex scaling, expanding understanding without relying on hyperbolic geometry.

Findings

Proves global rigidity using elementary methods

Avoids complex hyperbolic geometry techniques

Utilizes eigenvalue continuity and convex functions

Abstract

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.