Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case

TL;DR

This paper presents a new proof of the infinitesimal rigidity of smooth convex surfaces with positive Gauss curvature, using derivatives of the Hilbert-Einstein functional, and explores dualities with volume-based approaches in different geometries.

Contribution

It introduces a novel approach to proving infinitesimal rigidity by analyzing the Hilbert-Einstein functional derivatives, connecting it to duality principles in various geometric spaces.

Findings

New proof of infinitesimal rigidity for smooth convex surfaces

Establishes duality between Hilbert-Einstein functional and volume

Suggests future research directions in hyperbolic 3-manifolds

Abstract

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives of the Hilbert-Einstein functional with boundary term. This approach is in a sense dual to proving the Gauss infinitesimal rigidity, that is rigidity with respect to the Gauss curvature parametrized by the Gauss map, by studying derivatives of the volume bounded by the surface. We recall that Blaschke's classical proof of the infinitesimal rigidity is also related to the Gauss infinitesimal rigidity, but in a different way: while Blaschke uses Gauss rigidity of the same surface, we use the Gauss rigidity of the polar dual. 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 indicate directions for future research, including the infinitesimal rigidity of convex cores of hyperbolic 3--manifolds.