Calculus of Variations and Geometric Measure Theory
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Some results on codimension one rectifiability for subsets in Carnot groups

Gioacchino Antonelli (Scuola Normale Superiore)

created by gelli on 16 Oct 2019

23 oct 2019 -- 17:00   [open in google calendar]

Sala Seminari (Dipartimento di Matematica di Pisa)

Abstract.

At the beginning of the seminar, we introduce the setting of Carnot groups and, in this realm, we discuss the link between different notions of rectifiability: the general one for metric spaces by H.Federer and the ones proposed in the specific case of Carnot groups by Pauls and Franchi-Serapioni-Serra Cassano. We focus mainly on the rectifiability for sets of codimension 1 and present some known results that higlight the relationship between those notions.

In the second part of the seminar we construct a smooth non-characteristic hypersurface in a Carnot group of topological dimension 8 that is purely Pauls' unrectifiable. This shows that even very regular objects of codimension 1 that for sure are rectifiable according to the definition of Franchi-Serapioni-Serra Cassano are not so according to Pauls' definition. Finally we discuss how to prove that such an example does not exist in the Heisenberg group $\mathbb H^n$, with $n\geq 2$. That is to say, every smooth hypersurface in $\mathbb H^n$, with $n\geq 2$, is Pauls' rectifiable, even with bi-Lipschitz maps.

This is a joint work with E. Le Donne.

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