RUMIN'S COMPLEX AND INTRINSIC GRAPHS IN CARNOT GROUPS

This thesis is concerned with some aspects of geometric analysis on Carnot groups. In the first chapter, we study differential forms and Rumin's complex on Carnot groups. In particular, we undertake the analysis of Rumin's Laplacian $\Delta_R$ on the Heisenberg group. We obtain a decomposition of the space of Rumin's forms with $L^2$ coefficients into invariant subspaces and describe the action of $\Delta_R$ restricted to these subspaces up to unitary equivalence. We also obtain that this decomposition provide a $L^p$ decomposition of the space of Rumin's forms. In the second chapter, we study intrinsic Lipschitz graphs and intrinsic differentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. In particular, we prove that one codimensional intrinsic Lipschitz graphs are sets with locally finite $\G$-perimeter. From this a Rademacher's type theorem for one codimensional graphs in a general class of groups is proved.