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The normal scalar curvature conjecture for submanifolds

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Let MnM^n be an nn-dimensional manifold isometrically immersed in the space form Nn+m(c)N^{n+m}(c) of constant sectional curvature cc. Let hh be the second fundamental form, let H=1n trace⁡hH=\frac 1n\,\operatorname{trace} h be the mean curvature tensor, and let ρ\rho and ρ⊥\rho^\perp denote the normalized scalar curvature and normalized scalar curvature of the normal bundle, respectively. Normal scalar curvature conjecture.

ρ+ρ⊥≤∣H∣2+c.\rho+\rho^\perp\leq |H|^2+c.

This is also known as the DDVV conjecture and is a fundamental curvature inequality in submanifold geometry. The paper states that the conjecture is proved in its first part.

References

Primary source

Zhiqin Lu, “Normal scalar curvature conjecture and its applications”, arXiv:0803.0502 (2011).

Additional references

2 papers in this index state this conjecture (2007–2008). The statement above is taken from the most recent of them; the others are arXiv:0711.3510.

curvature-inequalitiessubmanifold-geometry
Posted

Progress summary

Refreshed
Claimed solved

The conjecture was proved by two independent mathematical teams, and no credible later challenge to the result was found.

De Smet, Dillen, Verstraelen, and Vrancken proposed the normal scalar curvature conjecture in 1999. It asserts the curvature inequality ρ+ρ⊥≤∣H∣2+c\rho+\rho^\perp\leq |H|^2+c for submanifolds of space forms.

Known results

  • Partial cases included n=2,3n=2,3 and m=2,3m=2,3, together with earlier special cases for m=2m=2.
  • The conjecture reduces to an algebraic inequality for arbitrary real symmetric matrices.
  • Equality cases were characterized through the corresponding shape operators.

Independent proofs (2007–2008)

Zhiqin Lu announced a proof in 2007, published in 2011, while Jianquan Ge and Zizhou Tang gave an independent proof published in 2008. The resulting theorem establishes ρ+ρ⊥≤∣H∣2+c\rho+\rho^\perp\leq |H|^2+c and also yields equality characterizations and further geometric applications.

Current status (as of August 2026): The DDVV conjecture is resolved, with independent published/preprint proofs and no recorded unresolved objection; its equality cases are also characterized.

Sources

Solutions 0

No solutions have been posted yet.