Abstract
We study biharmonic hypersurfaces in the space forms \(\overline{M}^{6}(c)\) with at most four distinct principal curvatures and whose second fundamental form is of constant norm. We prove that every such biharmonic hypersurface in \(\overline{M}^{6}(c)\) has constant mean curvature and constant scalar curvature. In particular, every such biharmonic hypersurface in \(\mathbb {S}^{6}(1)\) has constant mean curvature and constant scalar curvature. Every such biharmonic hypersurface in Euclidean space \(E^6\) and in hyperbolic space \(\mathbb {H}^{6}\) must be minimal and have constant scalar curvature.
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Gupta, R.S. On biharmonic hypersurfaces in 6-dimensional space forms. Afr. Mat. 30, 1205–1221 (2019). https://doi.org/10.1007/s13370-019-00714-y
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DOI: https://doi.org/10.1007/s13370-019-00714-y
Keywords
- Chen’s conjecture
- Generalized Chen’s conjecture
- Balmus–Montaldo–Oniciuc conjecture
- Mean curvature vector