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

Abstract

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