Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case

TL;DR

This paper introduces a new proof of the infinitesimal rigidity of convex polyhedra using derivatives of the discrete Hilbert-Einstein functional, offering a dual perspective to volume-based methods and exploring implications in spherical and hyperbolic geometries.

Contribution

It provides a novel approach to proving infinitesimal rigidity of convex polyhedra via the second derivative of the Hilbert-Einstein functional, extending the theory to spherical and hyperbolic spaces.

Findings

New proof of infinitesimal rigidity for convex polyhedra

Duality between Hilbert-Einstein functional and volume in different geometries

Potential for elementary proofs in hyperbolic and cone-manifolds

Abstract

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra, which deals with deformations that preserve face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity. We also discuss directions for future research, including elementary proofs of the infinitesimal rigidity of hyperbolic (cone-)manifolds and development of a discrete Bochner technique.