Differential forms in Carnot groups: a Γ-convergence approach

Abstract

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E * 0 , d c ) of "intrinsic" differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials d c appear naturally as limits of usual "Riemannian" differentials d ε , ε > 0. More precisely, we show that L 2 -energies associated with ε −κ d ε on 1-forms -converge, as ε → 0, to the energy associated with d c .

Sharp a priori estimates for div–curl systems in Heisenberg groups

“…In the last few years, there have been several instances that show the "naturalness" of using the Rumin complex instead of de Rham complex for Carnot groups. For example, we can quote [FT12], Theorem 3.16, where the authors study the naturalness of d c in terms of homogeneous homomorphisms of the group G, or [BF12] where the authors show that d c appears, in the spirit of the Riemannian approximation, as limit.…”

Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups

“…The first addend is an increasing function of t, while the second one is an absolutely continuous function of t. Therefore, by integrating the differential inequality (28), we get that…”

“…In this section we study the integrals A(t) and B 2 (t) appearing in the right-hand side of the monotonicity formula (28).…”

Variational formulas for submanifolds of fixed degree

“…Their mass coincides with our area (3.7) on intrinsic C 1 submanifolds. However in (3.8) we consider all possible m-forms and not only the intrinsic m-forms in the Rumin's complex [1,42,49].…”