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

The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo [18], who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo’s conjecture was solved by Bobenko-Pinkall-Springborn [3], 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.