  In this paper we solve a problem raised by Guti\'errez and Montanari about
comparison principles for $H-$convex functions on subdomains of Heisenberg
groups. Our approach is based on the notion of the sub-Riemannian horizontal
normal mapping and uses degree theory for set-valued maps. The statement of the
comparison principle combined with a Harnack inequality is applied to prove the
Aleksandrov-type maximum principle, describing the correct boundary behavior of
continuous $H-$convex functions vanishing at the boundary of horizontally
bounded subdomains of Heisenberg groups. This result answers a question by
Garofalo and Tournier. The sharpness of our results are illustrated by
examples.

Analysis of PDEs