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Veronese Projective Plane

The Veronese projective plane is the minimal surface in the unit hypersphere S^4 obtained from the real projective plane by

F([u])=sqrt(3/2)(uu^T-1/3I),

where u in S^2 is a unit column vector, [u]=[-u], and the ambient five-dimensional space of traceless real symmetric 3×3 matrices has inner product <A,B>=tr(AB) (Bryant 1982, Carlotto et al. 2026). Its induced metric has Gaussian curvature 1/3 and area 6pi. It is related to the projective Veronese surface, with this specified spherical normalization.

Carlotto et al. (2026) proved that every closed connected smooth embedded minimal surface in unit S^4 that is not a great 2-sphere has Morse index at least 5, with equality iff it is congruent to the Veronese projective plane. The Morse index counts independent negative directions of the area second variation. The two-to-one parametrization by S^2 has index 10, rather than 5. Carlotto et al. (2026) report AI assistance with the mathematical proof.

See also

Minimal Surface, Real Projective Plane, Veronese Surface

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References

Bryant, R. L. "Conformal and Minimal Immersions of Compact Surfaces into the 4-Sphere." J. Differential Geom. 17, 455-473, 1982. https://doi.org/10.4310/jdg/1214437137.Carlotto, A.; Li, Y.; and Wang, Z. "Morse Index Characterization of the Veronese Projective Plane." 26 Oct 2026. https://arxiv.org/abs/2610.35820.

Cite this as:

Weisstein, Eric W. "Veronese Projective Plane." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VeroneseProjectivePlane.html

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