$L_r$-biharmonic hypersurfaces in $\mathbb{E}^4$

Abstract

A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is said to be biharmonic if $\Delta^2x=0$, where $\Delta$ is the Laplace operator of $M^n$. Based on a well-known conjecture of Bang-Yen Chen, the only biharmonic hypersurfaces in $E^{n+1}$ are the minimal ones. In this paper, we study an extension of biharmonic hypersurfaces in 4-dimentional Euclidean space $\mathbb{E}^4$. A hypersurface $x : M^n\rightarrow\mathbb{E}^{n+1}$ is called $L_r$-biharmonic if $L_r^2x=0$, where $L_r$ is the linearized opereator of $(r + 1)$th mean curvature of $M^n$. Since $L_0=\Delta$, the subject of $L_r$-biharmonic hypersurface is an extension of biharmonic ones. We prove that any $L_2$-biharmonic hypersurface in $\mathbb{E}^4$ with constant $2$-th mean curvature is $2$-minimal. We also prove that any $L_1$-biharmonic hypersurfaces in $\mathbb{E}^4$ with constant mean curvature is $1$-minimal.