2010 # Biharmonic hypersurfaces in 4‐dimensional space forms

**Abstract:** We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms

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“…Biharmonic submanifolds have been studied by many geometers. For example, see [2], [3], [7], [8], [11], [12], [13], [14], [15], [18], [20], [21], [22], and the references therein. In a different setting, in [9], Chen defined a biharmonic submanifold M ⊂ E n of the Euclidean space as its mean curvature vector field H satisfies ∆H = 0 , where ∆ is the Laplacian.…”

confidence: 99%

“…Biharmonic submanifolds have been studied by many geometers. For example, see [2], [3], [7], [8], [11], [12], [13], [14], [15], [18], [20], [21], [22], and the references therein. In a different setting, in [9], Chen defined a biharmonic submanifold M ⊂ E n of the Euclidean space as its mean curvature vector field H satisfies ∆H = 0 , where ∆ is the Laplacian.…”

confidence: 99%

“…Then, using certain results on isoparametric hypersurfaces, it was proved that there exist no compact properbiharmonic hypersurfaces of constant mean curvature and with three distinct principal curvatures everywhere in the unit Euclidean sphere (see [9]). …”

confidence: 99%

“…(b) Any biharmonic hypersurfaces in H 4 (−1) is minimal (cf. [4]). (c) Any weakly convex biharmonic hypersurfaces in space form N m+1 (c) with c ≤ 0 is minimal (cf.…”

confidence: 99%