Inserted: 7 may 2008
Last Updated: 3 may 2011
Journal: Publ. Mat.
We prove that, in general, intrinsicly regular surfaces in the Heisenberg group $\mathbb H^1$ are not biLipschitz equivalent to the plane $\mathbb R^2$ with the ``parabolic'' distance, which instead models $C^1$ surfaces as proved by D. R. Cole and S. D. Pauls. In Heisenberg groups $\mathbb H^n$, the former surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps, thus answering a question raised by B. Kirchheim and F. Serra Cassano.
Keywords: Heisenberg group, parametrizations