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Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3
Tohoku Math. J.
67 (2015), 465479

Title
BIHARMONIC HYPERSURFACES WITH THREE DISTINCT PRINCIPAL CURVATURES IN EUCLIDEAN SPACE
Author
Yu Fu
(Received February 21, 2014, revised June 24, 2014) 
Abstract.
The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([1013, 16, 1821, 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 ChenMunteanu [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 abovementioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.
Mathematics Subject Classification.
Primary 53D12; Secondary 53C40.
Key words and phrases.
Chen's conjecture, biharmonic submanifolds.


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