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arXiv · 1911.02625

Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces

Stefano Montaldo·Alvaro Pampano
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Abstract

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only non-trivial example of a totally biharmonic surface appears in the product space $\mathbb{S}^2(\rho)\times\mathbb{R}$.

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Stefano Montaldo, Alvaro Pampano. 2019-11-06. Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces. https://arxiv.org/abs/1911.02625

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