Conformal-biharmonic hypersurfaces in spheres and product spaces

TL;DR

This paper studies conformal-biharmonic hypersurfaces in spheres and product spaces, classifying those with constant principal curvatures and scalar curvature, and exploring their geometric properties and global behavior.

Contribution

It provides a classification of conformal-biharmonic hypersurfaces with constant principal curvatures and scalar curvature in spheres and product spaces, including new global and structural results.

Findings

Hypersurfaces are either totally geodesic or cylindrical in product spaces.

Complete classification of isoparametric c-biharmonic hypersurfaces in spheres.

Totally umbilical c-biharmonic hypersurfaces are necessarily totally geodesic.

Abstract

The conformal-bienergy functional $E_2^c$ is a modified version of the classical bienergy functional $E_2$ and it is conformally invariant in the case of a four-dimensional domain. The critical points of $E_2^c$ are called conformal-biharmonic and denoted $c$-biharmonic. In the first part of the paper we study the $c$-biharmonic hypersurfaces $M^m$ with constant principal curvatures in the product space $ {\mathbb L}^m(\varepsilon) \times \mathbb{R} $, where $ {\mathbb L}^m(\varepsilon) $ denotes a space form of constant sectional curvature $ \varepsilon $. Specifically, we demonstrate that $ M^m $ is either totally geodesic or a cylindrical hypersurface of the form $ M^{m-1} \times \mathbb{R} $, where $ M^{m-1} $ is an iso\-parametric $c$-biharmonic hypersurface in $ {\mathbb L}^m(\varepsilon) $. In the second part of this article we obtain a full description of isoparametric $c$-biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ and a complete classification of $c$-biharmonic hypersurfaces with constant scalar curvature in $\mathbb{S}^{m+1}$, $m=2,3$ and $m=4$ with an additional assumption. In this context, we shall also prove a global result for compact $c$-biharmonic immersions in $\mathbb{S}^5$. In the final part of the paper, as a preliminary effort to understand $c$-biharmonic hypersurfaces in $ {\mathbb L}^m(\varepsilon) \times \mathbb{R} $ with \textit{non-constant} mean curvature, we establish that a totally umbilical $c$-biharmonic hypersurface must necessarily be totally geodesic.