OAI.YauCounterexamples.sphere_two_torus_two

The theorem states that there exists a smooth (C^∞) positive-definite Riemannian metric g on the four-manifold S² × S¹ × S¹ (the product of the unit 2-sphere in ℝ³ with a two-torus, modeled on ℝ² × ℝ × ℝ) such that g has unbounded nodal ratio in dimension d = 4. This means there are a sequence of smooth, not identically zero functions u_j and positive eigenvalues λ_j for the Laplace–Beltrami operator of this fixed metric g, computed from the local coordinate formula, with −Δ_g u_j = λ_j u_j everywhere, such that each zero set {u_j = 0} has finite 3-dimensional Hausdorff measure with respect to the path distance of g itself. Moreover λ_j tends to infinity and the ratio of the 3-dimensional nodal measure of u_j to √λ_j also tends to infinity as j → ∞. The statement is an admitted theorem in the source, not a verified proof.