A characterization of submanifolds of R^mxn satisfying optimal rigidity estimates

By a well-known theorem of Friesecke, James and Müller from 2002, the set of rotations SO(n) satisfies the following optimal rigidity estimate: If u is an H^1-vector field defined on some bounded Lipschitz domain, then the L^2-distance of Du to a single rotation can be controlled by the L^2-norm of the distance of Du to SO(n). We are interested in the following question: Which structural properties of the set SO(n) enable it to satisfy the aforementioned rigidity estimate? The case of smooth connected submanifolds of R^2x2 has recently been settled by Lamy, Lorent and Peng. We will discuss a new approach to the problem, which allows one to prove necessary and sufficient conditions for a general compact C^1-submanifold of R^mxn to satisfy such a rigidity estimate. Furthermore, this approach allows one to show that the validity of this estimate is stable under small graphical perturbations of the set K. This talk is based on my master's thesis, written under the supervision of Konstantinos Zemas and Sergio Conti.