Pansu Differentiability of Lipschitz Mappings Between Carnot Groups I

In this talk, I will present a brief introduction to Carnot groups (which, as you all know, include the class of Heisenberg groups) and the Carnot-Caratheodory metric spaces and then define the notion of Pansu derivative of a function between Carnot groups. Pansu-Rademacher differentiability theorem states that Lipschitz mappings between Carnot groups are Pansu differentiable almost everywhere. I will prove this result and try to be as detailed oriented as possible. Not surprisingly, the proof is similar to the proof of the classical Rademacher theorem but there are some technical difficulties in the way due to the rather complicated structure of Carnot groups. I'll conclude this talk by proving an interesting consequence of the Pansu-Rademacher theorem which shows that there is no biLipschitz embedding of any Heisenberg group (actually, any noncommutative Carnot group) into any Euclidean space. It is worth mentioning that, except for one step of the proof, I will follow the original proof of the Pansu-Rademacher theorem due to Pierre Pansu.