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

TL;DR

This paper classifies totally biharmonic hypersurfaces in space forms and 3D BCV spaces, showing they are isoparametric and providing explicit classifications, with a unique example in product spaces.

Contribution

It establishes that totally biharmonic hypersurfaces are isoparametric and fully classifies them in space forms and BCV spaces, including the unique non-trivial example in product spaces.

Findings

Totally biharmonic hypersurfaces are isoparametric in space forms.

Complete classification of such hypersurfaces in space forms and BCV spaces.

Identification of a unique non-trivial totally biharmonic surface in () imes.

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 $\mathbb{S}^2(\rho)\times\mathbb{R}$.