  We prove an almost monotonicity formula for H-minimal Legendrian Surfaces
(also called {\it contact stationary legendrian immersions}) in the Heisenberg
Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type
theorem for H-minimal Legendrian Surfaces. This last result happens to be in
particular a central ingredient in the analysis part of the proof of the
Willmore conjecture given by the author in arXiv:2007.05467
