Mathematics – Differential Geometry
Scientific paper
2011-05-25
Mathematics
Differential Geometry
60 pages
Scientific paper
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.
No associations
LandOfFree
Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional II: Smooth case will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-191794