  We describe three elementary models in three dimensional subelliptic geometry
which correspond to the three models of the Riemannian geometry (spheres,
Euclidean spaces and Hyperbolic spaces) which are respectively the SU(2),
Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau
inequalities on positive solutions of the heat equation. We use for that the
$\Gamma_{2}$ techniques that we adapt to those elementary model spaces. The
important feature developed here is that although the usual notion of Ricci
curvature is meaningless (or more precisely leads to bounds of the form
$-\infty$ for the Ricci curvature), we describe a parameter $\rho$ which plays
the same role as the lower bound on the Ricci curvature, and from which one
deduces the same kind of results as one does in Riemannian geometry, like heat
kernel upper bounds, Sobolev inequalities and diameter estimates.
