Tohoku Mathematical Journal

Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

Yu Fu

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Abstract

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10--13, 16, 18--21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvature ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.

Article information

Source
Tohoku Math. J. (2), Volume 67, Number 3 (2015), 465-479.

Dates
First available in Project Euclid: 6 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1446818561

Digital Object Identifier
doi:10.2748/tmj/1446818561

Mathematical Reviews number (MathSciNet)
MR3420554

Zentralblatt MATH identifier
1283.53005

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53C40: Global submanifolds [See also 53B25]

Keywords
Chen's conjecture biharmonic submanifolds

Citation

Fu, Yu. Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space. Tohoku Math. J. (2) 67 (2015), no. 3, 465--479. doi:10.2748/tmj/1446818561. https://projecteuclid.org/euclid.tmj/1446818561


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