Preprint

Inserted: 11 jan 2024
Last Updated: 11 jan 2024

Year: 2024

Abstract:

In this paper we achieve a first concrete step towards a better understanding of the so-called Bernstein problem in higher dimensional Heisenberg groups. Indeed, in the sub-Riemannian Heisenberg group $\mathbb{H}^n$, with $n\geq 2$, we show that the only entire hypersurfaces with vanishing horizontal symmetric second fundamental form and countable characteristic set are hyperplanes. This result relies on a sub-Riemannian characterization of a higher dimensional ruling property, as well as on the study of sub-Riemannian geodesics on Heisenberg hypersurfaces.

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