Differential forms in Carnot groups: a Γ-convergence approach

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 $\kappa$, intrinsic 1-forms as well as their intrinsic differentials $d_c$ appear naturally as limits of usual ``Riemannian'' differentials $d_\eps$, $\eps>0$. More precisely, we show that $L^2$-energies associated with $\eps^{-\kappa}d_\eps$ on 1-forms $\Gamma$-converge, as $\eps\to 0$, to the energy associated with $d_c$.