2015 # On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb S^5$

**Abstract:** We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere S 5 must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E 5 or hyperbolic space H 5 , which give affirmative partial answers to Chen's conjecture and Generalized Chen's conjecture.

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“…We first recall some relations concerning the coefficients of connection and principal curvature functions verified in [20] or [19]. Actually, they are the four dimension version of Lemmas 3.5 and 3.6 in [20] while Lemma 3.4 holds.…”

confidence: 99%

“…We first recall some relations concerning the coefficients of connection and principal curvature functions verified in [20] or [19]. Actually, they are the four dimension version of Lemmas 3.5 and 3.6 in [20] while Lemma 3.4 holds.…”

confidence: 99%

“…Then, we are able to prove that a biharmonic hypersurface with constant scalar curvatures in a space form M n (c) must have constant mean curvature, provided that the number of distinct principal curvature is no more than six. We would like to point out that our approach in this paper is different from those in [21], [22], [13], [5].…”

confidence: 97%

“…Some estimate for scalar curvature of compact proper biharmonic hypersurfaces with constant scalar curvature in spheres was obtained in [4]. Recently, it was proved in [22] that a biharmonic hypersurface with constant scalar curvature in the 5-dimensional space forms M 5 (c) necessarily has constant mean curvature.…”

confidence: 99%

“…It was proved in [5] (see also [2]) that form m ≥ 2, if a compact biharmonic hypersurface in sphere S m+1 (1) with the squared norm of the second fundamental form satisfies |A| 2 ≤ m, then |A| 2 = 0, or |A| 2 = m and it has constant mean curvature. Also, in a recent work [8], Fu proved that a biharmonic hypersurface with constant scalar curvature in 5-dimensional space forms M 5 (C) has constant mean curvature, and later in [9], it was proved that a biharmonic hypersurface with constant scalar curvature and at most six distinct principal curvatures in space forms M m+1 (C) has constant mean curvature. Our next theorem shows that the same result in [9] holds when we replace the principal curvature assumption by the compactness of the hypersurface.…”

confidence: 98%