Biharmonic Hypersurfaces in Euclidean Spaces

TL;DR

This paper investigates biharmonic hypersurfaces in Euclidean spaces, establishing that holonomic biharmonic hypersurfaces are minimal, thereby contributing to the classification of such geometric objects.

Contribution

It proves that holonomic biharmonic hypersurfaces in Euclidean spaces are necessarily minimal, providing a new characterization and construction method for these hypersurfaces.

Findings

Holonomic biharmonic hypersurfaces are minimal.

Construction methods for biharmonic hypersurfaces are discussed.

The study extends understanding of biharmonic and biconservative hypersurfaces.

Abstract

An isometric immersion $X: \Sigma^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic if $\Delta^2 X = 0$, i.e. if $\Delta H =0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of $\Sigma^n$ respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that $\Sigma^n$ is a holonomic biharmonic hypersurface iff it is minimal.