Abstract
We prove that, in general, ${\mathbb H}$-regular surfaces in the Heisenberg group $\mathbb{H}^1$ are not bi-Lipschitz equivalent to the plane ${\mathbb R}^2$ endowed with the ``parabolic'' distance, which instead is the model space for $C^1$ surfaces without characteristic points. In Heisenberg groups $\mathbb{H}^n$, ${\mathbb H}$-regular surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps.
Citation
Francesco Bigolin. Davide Vittone. "Some Remarks About Parametrizations of Intrinsic Regular Surfaces in the Heisenberg Group." Publ. Mat. 54 (1) 159 - 172, 2010.
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