Some remarks about parametrizations of intrinsic regular surfaces in the Heisenberg group

We prove that, in general, H-regular surfaces in the Heisenberg group H^1 are not bi-Lipschitz equivalent to the plane R^2 endowed with the \u201cparabolic\u201d distance, which instead is the model space for C^1 surfaces without characteristic points. In Heisenberg groups H^n, H-regular surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps