The graphs of Lipschitz functions and minimal surfaces on Carnot groups

We study and solve a new problem for the class of Lipschitz mappings (with respect to sub-Riemannian metrics) on Carnot groups. We introduce the new concept of graph for the functions on a Carnot group, and then the new concept of sub-Riemannian differentiability generalizing $hc$-differentiability. We prove that the mapping-“graphs” are almost everywhere differentiable in the new sense. For these mappings we define a concept of intrinsic measure and obtain an area formula for calculating this measure. By way of application, we find necessary and sufficient conditions on the class of surface-“graphs” under which they are minimal surfaces (with respect to the intrinsic measure of a surface).