Analysis Seminar

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Rectifiability in Carnot groups

Speaker: Gioacchino Antonelli, Courant Institute

Location: Warren Weaver Hall 1302

Date: Thursday, November 10, 2022, 11 a.m.

Synopsis:

The talk will be divided into two parts.

In the first introductory part I will recall some of the main results in the classical theory of rectifiability in Euclidean spaces, such as Marstrand—Mattila’s rectifiability criterion, and Preiss’ Theorem. I will mention some recent extensions in metric spaces. Moreover, I will discuss basic properties and definitions of Carnot groups.
In the second part I will describe some attempts to give a flexible notion of regular surface and rectifiability in the setting of Carnot groups. After recalling some “local” definitions of rectifiability, I shall focus on the “infinitesimal” notion of P-rectifiable measure recently introduced by A. Merlo, and studied by myself and A. Merlo.

A P-rectifiable measure of dimension h in a Carnot group is a Radon measure with (almost everywhere) positive lower h-density, finite upper h-density, and such that the tangent measures are Haar measures on homogeneous subgroups of homogeneous dimension h of the ambient Carnot group.

I will present an equivalent characterization of P-rectifiability by means of coverings with intrinsic differentiable graphs in the case the tangents are complemented.
Finally, I shall discuss an extension of Marstrand—Mattila's rectifiability criterion for P-rectifiable measures in the case the tangents are complemented by a normal homogeneous subgroup, and how this implies an extension of Preiss’ Theorem for 1-dimensional measures in the first Heisenberg group with the Koranyi distance.

The results presented are obtained in collaboration with E. Le Donne and A. Merlo.