Conformal-biharmonic hypersurfaces in spheres and product spaces

TL;DR

This work studies the conformal-bienergy functional $E_2^c$ and its $c$-biharmonic hypersurfaces, with emphasis on ${\mathbb L}^m(\\varepsilon)\\times{\mathbb R}$ and ${\mathbb S}^{m+1}$. It develops the $c$-biharmonic tension field, analyzes constant principal curvatures, and conducts a detailed isoparametric analysis in spheres, obtaining complete classifications for several degrees $\\ell=1,2,3,4,6$. The results include that minimal isoparametric $c$-biharmonic hypersurfaces occur for $\\ell=3,6$, and that for degree $\\ell=4$ there exist non-minimal examples when $m_1<m_2$, along with a global rigidity and a uniqueness statement in various low dimensions. These contributions provide explicit classifications and rigidity phenomena for conformally invariant variational problems in differential geometry, with potential implications for stability and geometric analysis in product and spherical ambient spaces.

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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Remark 2.2