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Some Remarks About Parametrizations of Intrinsic Regular Surfaces in the Heisenberg Group

Francesco Bigolin and Davide Vittone

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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.

Article information

Publ. Mat., Volume 54, Number 1 (2010), 159-172.

First available in Project Euclid: 8 January 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C17: Sub-Riemannian geometry 54E40: Special maps on metric spaces

Bi-Lipschitz parametrizations ${\mathbb H}$-regular surfaces Heisenberg group


Bigolin, Francesco; Vittone, Davide. Some Remarks About Parametrizations of Intrinsic Regular Surfaces in the Heisenberg Group. Publ. Mat. 54 (2010), no. 1, 159--172.

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