Davide Vittone: Differentiability of intrinsic Lipschitz graphs in Carnot groups

"Sub-Riemannian seminar" of December 11, 2020.

Speaker: Davide Vittone

Title: Differentiability of intrinsic Lipschitz graphs in Carnot groups

Abstract:
Submanifolds with intrinsic Lipschitz regularity in sub-Riemannian
Carnot groups can be introduced using the theory of intrinsic
Lipschitz graphs started by B. Franchi, R. Serapioni and F. Serra
Cassano almost 15 years ago. One of the main related questions
concerns a Rademacher-type theorem (i.e., existence of a tangent
plane) for such graphs: in this talk I will discuss a recent positive
solution to the problem in Heisenberg groups. The proof uses currents
in Heisenberg groups (in particular, a version of the celebrated
Constancy Theorem) and a number of complementary results such as
extension and smooth approximation theorems for intrinsic Lipschitz
graphs. I will also show a recent example (joint with A. Julia and S.
Nicolussi Golo) of an intrinsic Lipschitz graph in a Carnot group that
is nowhere intrinsically differentiable.