Geometric rigidity estimates on the sphere

In this talk I would like to present some connections between the isoperimetric inequality, the stability of Möbius transformations of the sphere and the well known geometric rigidity estimates of Friesecke, James and Müller regarding the stability of the special orthogonal group. The main result is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation of the sphere, an average conformal-isoperimetric type of deficit controls the deviation (in an average sense) of the map in question from a particular Möbius transformation. Its link to the geometric rigidity of SO(n) is subsequently discussed. This is joint work with Prof. Stephan Luckhaus.