## About the Project

### Brief Description

The core of this project can be shortly (and roughly) described as project in Geometric Metric Theory and curvature equations in non-Euclidean structures. It is worthwhile from the very beginning to state clearly that, when we mention non-Euclidean structures, we refer to metric structures that are not Euclidean at any scale. Thus, the model we have in mind are not Riemannian manifolds, but better the so-called sub-Riemannian manifolds and fractals, or even fractals in sub-Riemannian spaces. In the last few years, sub-Riemannian structures have been largely studied in several respects, such as differential geometry, geometric measure theory, subelliptic differential equations, complex variables, optimal control theory, mathematical models in neurosciences, non-holonomic mechanics, robotics.

Among all sub-Riemannian structures, a prominent position is taken by the so-called Carnot groups (simply connected Lie groups G with stratified nilpotent algebra), which play versus sub Riemannian spaces the role played by Euclidean spaces (considered as tangent spaces) versus Riemannian manifolds. The notion of dimension is crucial in our approach: the non-Euclidean character of the structures we are interested to study hides in the gap between the topological dimension of a group G and its metric dimension. Isoperimetric inequalities, analysis on fractal sets, quasiconformal and quasiregular maps are a typical manifestations of the metric dimension versus the topological dimension. In addition, dimension phenomena appear in a crucial way when dealing with intrinsic curvature in submanifolds of Carnot groups and in the curvature equations.

### Research Activity (Work Packages)

#### WP 1: Geometric Measure Theory

**Objectives**

Geometric measure theory is intimately connected with the calculus of variations, rectifiability,
potential theory and fractal geometry. Our work encompasses aspects of these subjects in the
setting of sub-Riemannian geometry and more general metric spaces. A deep understanding
of the structure of nonsmooth or fractal sets via the methods of geometric measure theory is
essential in the constructive aspects of mapping theory and geometric pde.

#### WP 2 : Mapping Theory

**Objectives**

A deep understanding of the properties of mappings in various classes clarifies the internal
geometric structure of the target and source spaces. In our research we focus on metric
properties measuring distortion of size and shape (quasiconformality, Lipschitz continuity) as
well as differential properties (Sobolev, harmonic and wave maps).

#### WP 3: Differential Form

**Objectives**

It is well known that differential forms on a manifold hides a large amount of information on the
topology of the manifold itself. In the last few years, an intrinsic complex of differential forms
has been associated in a natural way with any Carnot group following an idea due to M.
Rumin. We want to carry on the study of these forms, in collection with quasiregular maps in
groups.

#### WP 4: Geometric PDE's

**Objectives**

The general objective of the work is to describe surfaces of the subriemannian space through
one of its most important geometric invariant: curvature. Precisely we will study mean
curvature equations in Lie groups and Levi curvature in CR manifolds.