Let M^m be a hypersurface of the unit sphere S^m+1. It is proper biharmonic if it is biharmonic but not minimal. The standard products are the inclusions S^m_1(1/√2)×S^m_2(1/√2)⊂S^m+1, where m_1+m_2=m and m_1≠ m_2. Classification conjecture. The only proper biharmonic hypersurfaces in S^m+1 are the open parts of hyperspheres S^m(1/√2) or of the standard products of spheres S^m_1(1/√2)×S^m_2(1/√2), with m_1+m_2=m and m_1≠ 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.

Classification conjecture for proper biharmonic hypersurfaces in spheres

About 17 years old · traced to 2009

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

\mathbb{S}^{m_1}(1/\sqrt{2})\times\mathbb{S}^{m_2}(1/\sqrt{2})\subset\mathbb{S}^{m+1},

where $m_1+m_2=m$ and $m_1\neq m_2$ .

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

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 1Other wordings

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

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

References

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.

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Classification conjecture for proper biharmonic hypersurfaces in spheres

The classification conjecture for proper-biharmonic hypersurfaces in spheres

Primary source

Additional references

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