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Minimal Clifford Hypersurface


A minimal Clifford hypersurface in the unit sphere S^(m+1) is a product

 C_(k,m-k)=S^k(sqrt(k/m))×S^(m-k)(sqrt((m-k)/m)),

where m>=2, 1<=k<m, and the two factors occupy orthogonal Euclidean coordinate subspaces. The displayed radii place the product in the unit sphere and make its mean curvature vanish. Its volume is

 |S^k||S^(m-k)|(k/m)^(k/2)((m-k)/m)^((m-k)/2),

where |S^j| denotes the volume of the unit j-sphere. For m=2, this is the Clifford torus in S^3, with both radii 1/sqrt(2) and area 2pi^2. The minimum of these product volumes is the proposed threshold in the Solomon-Yau least-volume conjecture.


See also

Clifford Torus, Hypersphere, Minimal Submanifold, Solomon-Yau Least-Volume Conjecture, Torus, Totally Geodesic Submanifold

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References

Ge, J. and Li, F. "Volume Gap for Minimal Submanifolds in Spheres." J. Geom. Anal. 36, Article No. 291, 2026. https://doi.org/10.1007/s12220-026-02551-5.

Cite this as:

Weisstein, Eric W. "Minimal Clifford Hypersurface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MinimalCliffordHypersurface.html

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