Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

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.

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“…M 3 is the 1-minimal hypersurface (see example 3.1 of [10]), and hence by using (3.1) and (3.2), we get that M 3 is L 1 -biharmonic in E 4 .…”

Section: Preliminariesmentioning

confidence: 93%

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

Mohammadpouri

Pashaei

2019

B Soc Paran Mat

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Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

Chen

Güler

Yaylı

et al. 2023

International Electronic Journal of Geometry

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The theory of finite type submanifolds was introduced by the first author in late 1970s and it has become a useful tool for investigation of submanifolds. Later, the first author and P. Piccinni extended the notion of finite type submanifolds to finite type maps of submanifolds; in particular, to submanifolds with finite type Gauss map. Since then, there have been rapid developments in the theory of finite type. The simplest finite type submanifolds and submanifolds with finite type Gauss maps are those which are of 1-type. The classes of such submanifolds constitute very special and interesting families in the finite type theory.

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Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

Cited by 2 publications

References 21 publications

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

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

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

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