Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group

Kirchheim, Bernd; Serra Cassano, Francesco

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 871-896.

We construct an intrinsic regular surface in the first Heisenberg group ${ℍ}^1\equiv {ℝ}^3$ equipped wiht its Carnot-Carathéodory metric which has euclidean Hausdorff dimension $2.5$. Moreover we prove that each intrinsic regular surface in this setting is a $2$-dimensional topological manifold admitting a $\frac{1}{2}$-Hölder continuous parameterization.

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