Abstract
We obtain several rigidity results for biharmonic submanifolds in \(\mathbb{S}^{n}\) with parallel normalized mean curvature vector fields. We classify biharmonic submanifolds in \(\mathbb{S}^{n}\) with parallel normalized mean curvature vector fields and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector fields in \(\mathbb{S}^{n}\).
Then we investigate, for (not necessarily compact) proper-biharmonic submanifolds in \(\mathbb{S}^{n}\), their type in the sense of B.-Y. Chen. We prove that (i) a proper-biharmonic submanifold in \(\mathbb{S}^{n}\) is of 1-type or 2-type if and only if it has constant mean curvature f=1 or f∈(0,1), respectively; and (ii) there are no proper-biharmonic 3-type submanifolds with parallel normalized mean curvature vector fields in \(\mathbb{S}^{n}\).
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The first author was supported by Grant POSDRU/89/1.5/S/49944, Romania. The second author was supported by Contributo d’Ateneo, University of Cagliari, Italy. The third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-RU-TE-2011-3-0108.
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Balmuş, A., Montaldo, S. & Oniciuc, C. Biharmonic PNMC submanifolds in spheres. Ark Mat 51, 197–221 (2013). https://doi.org/10.1007/s11512-012-0169-5
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DOI: https://doi.org/10.1007/s11512-012-0169-5