An area formula in metric spaces

We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence of a measure theoretic notion of Jacobian, defined as the density of a suitable "pull-back measure"

Towards a theory of area in homogeneous groups

A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the homogeneous tangent space, that is a suitable weighted algebraic expansion of the submanifold. This space plays a central role for the existence of blow-ups. Main applications are area-type formulae for new classes of $C^1$ smooth submanifolds. We also study various classes of distances, showing how their symmetries lead to simpler area and coarea formulas. Finally, we establish the equality between spherical measure and Hausdorff measure on all horizontal submanifolds.Comment: 60 page

Study of Crystal-field Effects in Rare-earth (RE) - Transition-metal Intermetallic Compounds and in RE-based Laser Crystals

Rare-earth (RE) based compounds and alloys are of great interest both for their fundamental physical properties and for applications. In order to tailor the required compounds for a specific task, one must be able to predict the energy level structure and transition intensities for any magnetic ion in any crystalline environment. The crystal-field (CF) analysis is one of the most powerful theoretical methods to deal with the physics of magnetic ions. In the present work, this technique is used to analyze peculiar physical properties of some materials employed in the production of new-generation solid-state laser and high-performance permanent magnets.Comment: 6 pages, 2 figures; extended abstract of PhD thesis (final version with updated references

Contact equations, Lipschitz extensions and isoperimetric inequalities

We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups.Comment: This version has additional references and a revisited introductio

Blow-up of regular submanifolds in Heisenberg groups and applications

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group

A new differentiation, shape of the unit ball and perimeter measure

We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent groups. This result leads us to an unexpected relationship between the area formula with respect to a distance and the profile of its corresponding unit ball.Comment: 17 page

Nonexistence of horizontal Sobolev surfaces in the Heisenberg group

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group \H^1. Here we answer a question raised in a paper by Z.M.Balogh, R.Hoefer-Isenegger, J.T.Tyson

Friezes of type D

In this article, we establish a link between the values of a frieze of type D and some values of a particular frieze of type A. This link allows us to compute, independently of each other, all the cluster variables in the cluster algebra associated with a quiver Q of type D