Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{mathcal S}} in the Heisenberg group mathbb H¹{{mathbb {H}}^1} and we prove that for points p along the center of mathbb H¹{{mathbb {H}}^1} the quantity fracs_H(S?B(p,r))r^Q-1{frac{sigma_H(mathcal Scap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of mathbb H¹{{mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.