Rumin's Differential Forms and L^2-Hodge Decomposition in Sub-Riemannian Contact Manifolds

The aim of this thesis is to prove a Hodge Decomposition Theorem for differential forms in L^2, defined in the setting of the so-called Rumin’s complex on sub-Riemannian contact manifolds. The first chapters are dedicated to the construction of Rumin’s complex in Carnot groups, and in particular, in Heisenberg group. In many issues of Analysis and Differential Geometry, the Rumin’s complex turns out to be more adapted to the non-isotropic structure of Carnot groups than the usual de Rham’s complex. For example, there are several results about Sobolev-Poincaré inequalities for differential forms that were proved using Rumin’s complex to obtain sharp estimates. Later, we study the Rumin’s complex in sub-Riemannian contact manifolds, which are locally contact-diffeomorphic to the Heisenberg group. More precisely, in contact manifolds with bounded geometry, we generalize a Sobolev-Gaffney type inequality that was recently proved, and we write an estimate of global maximal hypoellipticity for Rumin’s Laplacian. Thanks to these results, using the variational approach that Morrey used for the Riemannian decomposition, we have proved not only the smooth Hodge Decomposition for Rumin’s complex on compact contact manifolds, but also the L^2 decomposition, obtaining regularity for the solution of Poisson’s equation for differential forms.