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arXiv · 2305.13148

A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups

Andrea Pinamonti·Simone Verzellesi
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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 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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Andrea Pinamonti, Simone Verzellesi. 2023-05-22. A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups. https://arxiv.org/abs/2305.13148

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