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Classification conjecture for proper biharmonic hypersurfaces in spheres

Let MmM^m be a hypersurface of the unit sphere Sm+1\mathbb{S}^{m+1}. It is proper biharmonic if it is biharmonic but not minimal. The standard products are the inclusions

Sm1(1/2)×Sm2(1/2)⊂Sm+1,\mathbb{S}^{m_1}(1/\sqrt{2})\times\mathbb{S}^{m_2}(1/\sqrt{2})\subset\mathbb{S}^{m+1},

where m1+m2=mm_1+m_2=m and m1≠m2m_1\neq m_2.

Classification conjecture. The only proper biharmonic hypersurfaces in Sm+1\mathbb{S}^{m+1} are the open parts of hyperspheres Sm(1/2)\mathbb{S}^m(1/\sqrt{2}) or of the standard products of spheres Sm1(1/2)×Sm2(1/2)\mathbb{S}^{m_1}(1/\sqrt{2})\times\mathbb{S}^{m_2}(1/\sqrt{2}), with m1+m2=mm_1+m_2=m and m1≠m2m_1\neq m_2.

The conjecture would classify all proper biharmonic hypersurfaces in spheres; the paper presents equivalent formulations involving principal curvatures, parallelness, constant mean curvature with non-negative sectional curvature, and isoparametricity. The general statement remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The classification conjecture for proper-biharmonic hypersurfaces in spheres

    Let n1,n2n_1,n_2 be positive integers satisfying n1+n2=n−1n_1+n_2=n-1 and n1≠n2n_1\neq n_2, and let Sn\mathbb S^n denote the unit Euclidean sphere. A hypersurface is proper-biharmonic if its inclusion map into Sn\mathbb S^n is biharmonic but non-minimal. The classification conjecture. The only proper-biharmonic hypersurfaces in Sn\mathbb S^n are the open parts of hyperspheres Sn−1(1/2)\mathbb S^{n-1}(1/\sqrt{2}) or of generalized Clifford tori

    Sn1(1/2)×Sn2(1/2).\mathbb S^{n_1}(1/\sqrt{2})\times \mathbb S^{n_2}(1/\sqrt{2}).

    The conjecture proposes a complete classification of proper-biharmonic hypersurfaces in spheres; the supplied text does not state whether it has been resolved.

    source: A. Balmuş and C. Oniciuc, “Biharmonic surfaces of S^4”, arXiv:0902.4849 (2009).

Sources & referencesView supporting material

Primary source

A. Balmus, S. Montaldo and C. Oniciuc, “New results toward the classification of Biharmonic submanifolds in S^n”, arXiv:1111.6063 (2012).

Additional references

2 papers in this index state this conjecture (2009–2011). The statement above is taken from the most recent of them; the others are arXiv:0908.3063.

biharmonic-geometrybiharmonic-submanifoldssphere-geometryspheressubmanifold-geometrysubmanifolds

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