OAI.YauCounterexamples.sphere_three

The theorem states that, for every smooth Riemannian metric gRound on the 3-sphere S³ (the unit sphere in ℝ⁴, modeled on Euclidean 3-space) that is round, meaning that at each point its inner product of tangent vectors v and w equals the Euclidean inner product of their images under the derivative of the inclusion S³ → ℝ⁴, the following holds. For every set N of smooth metrics on S³ that is a smooth neighborhood of gRound, there is a metric g in N that has unbounded nodal ratio in dimension 3. Here N is a smooth neighborhood of gRound if there are finitely many compact-chart tests, each specifying a chart center, a compact subset of that chart's target, a derivative order, and a row and column index, together with a single tolerance ε>0, such that every smooth metric g belongs to N whenever, for every test, the iterated derivative of that order of the difference between the chart coefficients g_{ij} and gRound_{ij} has norm below ε at all points of the compact set. A metric g has unbounded nodal ratio if there are smooth nonzero functions u_j on S³ and positive eigenvalues λ_j with -Δ_g u_j = λ_j u_j, where Δ_g is the Laplace–Beltrami operator given by the local coordinate formula, such that each zero set {u_j=0} has finite 2-dimensional Hausdorff measure for the path distance of g, the eigenvalues λ_j tend to infinity, and the ratio of that nodal measure to √λ_j also tends to infinity.