Towards a quantitative rigidity estimate for deformations in the hyperbolic space

The rigidity result of Friesecke-James-Mueller asserts that given any deformation in the Euclidean space of dimension n, the L2 distance of its gradient from a single rotation matrix is bounded by a multiple of the L2 distance from the group SO(n) of rotations.

We will present a program to adapt the result and the method of proof to the hyperbolic geometry. We will present the expected result in the hyperboloid model, which we will describe for our purpose. Some progress and technical difficulties will be discussed in detail.