  In this work, we give a survey on non characteristic domains of Heisenberg
groups. We prove that bounded domains which are diffeomorphic to the solid
torus having the center of the group as rotation axis, are non characteristic.
Then, we state the following conjecture : The bounded non characteristic
domains of the Heisenberg group of dimension 1 are those diffeomorphic to a
solid torus having the center of the group as rotation axis.
