Infinitesimal rigidity of smooth convex surfaces through the second derivative of the Hilbert-Einstein functional

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 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 infinitesimal rigidity is also related to Gauss infinitesimal rigidity, but in a different way: while Blaschke uses the Gauss rigidity of the same surface, we use the Gauss rigidity of the polar dual. The two connections between metric and Gauss deformations generate the Darboux wreath of 12 surfaces. The duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity, becomes perfect in the spherical and in the hyperbolic-de Sitter space.