Abstract:
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).
Keywords:
Carnot group, Lipschitz mapping, graph, area formula, minimal surface.
Citation:
M. B. Karmanova, “The graphs of Lipschitz functions and minimal surfaces on Carnot groups”, Sibirsk. Mat. Zh., 53:4 (2012), 839–861; Siberian Math. J., 53:4 (2012), 672–690