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 [17], 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 between vertex scaling, the geometry of polyhedra in $3$-dimensional hyperbolic space and the concavity of the volume of ideal and hyperideal 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 and the extension of convex functions.

Keywords

rigidity, vertex scaling, piecewise linear metric, piecewise hyperbolic metric

2010 Mathematics Subject Classification

52C25, 52C26

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The research of the second-named author is supported by the Fundamental Research Funds for the Central Universities under grant no. 2042020kf0199, and by the National Natural Science Foundation of China under grant no. 61772379.

Received 26 November 2020

Accepted 5 July 2021

Published 24 October 2022