[Paper Review] Proof of the normal scalar curvature conjecture

Zhiqin Lu|ArXiv.org|Nov 22, 2007

Geometric Analysis and Curvature Flows7 references18 citations

TL;DR

This paper proves the normal scalar curvature conjecture (also known as the DDVV conjecture) in submanifold geometry, establishing that for any isometric immersion of an n-dimensional manifold into an (n+m)-dimensional space form of constant curvature c, the sum of the normalized scalar curvatures of the tangent and normal bundles is bounded above by the square of the mean curvature plus c. The proof reduces the conjecture to a purely algebraic inequality involving symmetric matrices and uses invariant theory, matrix commutator estimates, and spectral analysis to establish the result for all n, m ≥ 1.

ABSTRACT

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

Motivation & Objective

• To resolve the long-standing normal scalar curvature conjecture (DDVV conjecture) in submanifold geometry.
• To establish a sharp inequality relating the intrinsic and normal scalar curvatures of an isometric submanifold in a space form.
• To prove that the conjecture holds universally for all dimensions n and codimensions m.
• To provide a complete solution using linear algebraic techniques, reducing the geometric problem to matrix inequalities.
• To settle the Böttcher-Wenzel conjecture as a byproduct, proving ||[X,Y]||² ≤ 2||X||²||Y||² for real matrices.

Proposed method

• Reduces the geometric conjecture to an equivalent algebraic inequality involving symmetric matrices A₁,…,Aₘ: (∑||Aᵣ||²)² ≥ 2∑||[Aᵣ,Aₛ]||² for r < s.
• Uses the invariance of the inequality under the action of O(n) × O(m) to simplify the configuration via orthogonal transformations.
• Applies a key lemma estimating weighted sums of squared differences of real numbers under zero-sum and unit-norm constraints.
• Employs spectral analysis and eigenvalue comparison techniques to bound the commutator terms.
• Uses singular value decomposition and orthogonal conjugation to normalize the matrix X with ||X|| = 1.
• Proves the Böttcher-Wenzel conjecture by analyzing the linear operator T(Y) = [Xᵀ,[X,Y]] and showing its largest eigenvalue is at most 2.

Experimental results

Research questions

• RQ1 Does the inequality ρ + ρ⊥ ≤ |H|² + c hold for all isometric submanifolds in space forms of constant curvature c?
• RQ2 Can the DDVV conjecture be reduced to a purely algebraic inequality involving symmetric matrices and their commutators?
• RQ3 Is the Böttcher-Wenzel conjecture ||[X,Y]||² ≤ 2||X||²||Y||² true for all real n×n matrices X and Y?
• RQ4 What is the sharp constant in the commutator norm inequality for real matrices?
• RQ5 Can the conjecture be proven uniformly across all dimensions n and codimensions m using invariant methods?

Key findings

• The normal scalar curvature conjecture is fully proven for all n, m ≥ 1, confirming that ρ + ρ⊥ ≤ |H|² + c holds universally.
• The equivalent matrix inequality (∑||Aᵣ||²)² ≥ 2∑||[Aᵣ,Aₛ]||² is established for all symmetric matrices Aᵣ and all n, m ≥ 1.
• The Böttcher-Wenzel conjecture is proven: ||[X,Y]||² ≤ 2||X||²||Y||² for all real n×n matrices X and Y.
• The maximum eigenvalue of the linear operator T(Y) = [Xᵀ,[X,Y]] is shown to be at most 2, confirming the conjecture.
• The proof uses invariant reduction to simplify the matrix configuration, assuming A₁ is diagonal and Aᵣ ⊥ Aₛ for r ≠ s.
• The result settles all previously known special cases and provides a uniform solution across all dimensions.