| import OAI.Geometry.NodalSets.Charts.SphereReferenceMeasurePositive |
| import OAI.Geometry.NodalSets.Spectral.SphereEigenGlobalRepresentative |
|
|
| namespace OAI |
|
|
| namespace Yau.Target |
| open MeasureTheory Manifold Set Yau.Geometry |
| open scoped ContDiff |
| noncomputable section |
| attribute [local instance] sphereGlobalRepresentativeMeasurable sphereGlobalRepresentativeBorel |
|
|
| theorem sphere_L2_representative_pairing (d : SphereEnergyData) (f : SphereWeightedL2 d) |
| (u : Base → ℝ) (ha : u =ᵐ[sphereWeightedMeasure d.density] (f : Base → ℝ)) : |
| sphereWeightedPairing d.density u u=‖f‖^2 := by |
| rw [sphereWeightedPairing,sphereWeightedL2_norm_sq_integral] |
| apply integral_congr_ae |
| filter_upwards [(sphereWeightedMeasure_ae_iff d _).mp ha] with x hx |
| rw [hx] |
| ring |
|
|
| theorem sphere_eigen_global_normalized (d : SphereEnergyData) |
| (hrho : ∀ p : Base, ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x))) |
| (mu : ℝ) (hmu : mu ≠ 0) (f : SphereWeightedL2 d) |
| (heigen : sphereL2Resolvent d f=mu • f) (hn : ‖f‖=1) : |
| ∃ u : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u ∧ |
| u =ᵐ[sphereWeightedMeasure d.density] (f : Base → ℝ) ∧ |
| sphereWeightedPairing d.density u u=1 ∧ u ≠ 0 ∧ |
| ∀ p x, x ∈ interior (Yau.realCenteredCube 4 (1/64)) → |
| Yau.coordDiv (realMatrixFlux (sphereChartPrincipalDensity d p) (u ∘ sphereChartCoordMap p)) x+ |
| sphereEigenForcingCoefficient d p mu x*u (sphereChartCoordMap p x)=0 := by |
| obtain ⟨u,hu,ha,he⟩ := sphere_eigen_global_representative d hrho mu hmu f heigen |
| have hnorm : sphereWeightedPairing d.density u u=1 := by |
| rw [sphere_L2_representative_pairing d f u ha,hn,one_pow] |
| refine ⟨u,hu,ha,hnorm,?_,he⟩ |
| intro hz |
| simp only [hz,sphereWeightedPairing,Pi.zero_apply,mul_zero,integral_zero] at hnorm |
| norm_num at hnorm |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
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|
