OAI.SharpNodal.Main

The theorem states that, for any compact connected metric space M that is a C^∞ manifold charted over the Euclidean plane ℝ², equipped with a C^∞ Riemannian metric on its tangent bundle that is compatible with the metric space structure (IsRiemannianManifold), the sharp nodal length bound MainStatement holds. Concretely, there is a constant C ≥ 0, depending on M, such that for every λ > 0 and every C^∞ function u : M → ℝ that is not identically zero and satisfies −Δu = λu at every point, the nodal length of u is at most C√λ. Here Δ is the Laplace–Beltrami operator, defined at x by the local coordinate formula in the chart at x: (1/√det g) Σᵢ ∂ᵢ(√det g Σⱼ g^{ij} ∂ⱼ(u∘chart⁻¹)), where g is the matrix of inner products of the Riemannian metric on the coordinate vector fields. The nodal set is {x : u(x)=0}, and its nodal length is its one-dimensional Hausdorff measure, taken in the metric of M with the Borel σ-algebra and valued in [0,∞]; the bound is stated as nodalLength(u) ≤ ENNReal.ofReal(C√λ). The theorem is admitted with sorry in the source.