Taiwanese Journal of Mathematics

$L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

M. Aminian and S. M. B. Kashani

Full-text: Open access

Abstract

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for $ L_k $-operators. The new conjecture ($ L_k $-conjecture) is formulated as follows: If $ L_k^{2}x=0 $ then $ H_{k+1}=0$ where $ x:M^{n}\rightarrow\Bbb{R}^{n+1} $ is an isometric immersion of a Riemannian manifold $ M^n $ into the Euclidean space $ \Bbb{R}^{n+1} $, $ H_{k+1} $ is the $ (k+1) $-th mean curvature of $ M $, and $ L_k $ is the linearized operator of the $ (k+1) $-th mean curvature of the Euclidean hypersurface $ M $. We prove the $ L_k $-conjecture for the hypersurface $ M $ with at most two principal curvatures.

Article information

Source
Taiwanese J. Math., Volume 19, Number 3 (2015), 861-874.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133666

Digital Object Identifier
doi:10.11650/tjm.19.2015.4830

Mathematical Reviews number (MathSciNet)
MR3353257

Zentralblatt MATH identifier
1357.53056

Subjects
Primary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
$L_k$-operator biharmonic Chen conjecture

Citation

Aminian, M.; Kashani, S. M. B. $L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE. Taiwanese J. Math. 19 (2015), no. 3, 861--874. doi:10.11650/tjm.19.2015.4830. https://projecteuclid.org/euclid.twjm/1499133666


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