| import OAI.Geometry.NodalSets.Spectral.SphereEigenMatrixEquation | |
| namespace OAI | |
| namespace Yau.Target | |
| open MeasureTheory Set Yau.Geometry | |
| open scoped ContDiff | |
| noncomputable section | |
| theorem sphere_eigen_rescaled_input (d : SphereEnergyData) (mu : ℝ) (hmu : mu ≠ 0) | |
| (f : SphereWeightedL2 d) (heigen : sphereL2Resolvent d f=mu • f) : | |
| sphereL2Resolvent d (mu⁻¹ • f)=f ∧ | |
| sphereL2Resolvent d (mu⁻¹ • f)=mu • (mu⁻¹ • f) := by | |
| have hr : sphereL2Resolvent d (mu⁻¹ • f)=f := by | |
| rw [map_smul,heigen,smul_smul,inv_mul_cancel₀ hmu,one_smul] | |
| refine ⟨hr,?_⟩ | |
| rw [hr,smul_smul,mul_inv_cancel₀ hmu,one_smul] | |
| theorem sphere_eigen_direct_matrix_equation (d : SphereEnergyData) (p : Base) | |
| (hrho : ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x))) | |
| (mu : ℝ) (hmu : mu ≠ 0) | |
| (f : SphereWeightedL2 d) (heigen : sphereL2Resolvent d f=mu • f) | |
| (v : Yau.Jets.Coord → ℝ) (hv : ContDiff ℝ ∞ v) | |
| (ha : v =ᵐ[volume.restrict (Yau.realCenteredCube 4 (1/32))] | |
| (fun x ↦ f (sphereChartCoordMap p x))) : | |
| ∀ x ∈ interior (Yau.realCenteredCube 4 (1/64)), | |
| Yau.coordDiv (realMatrixFlux (sphereChartPrincipalDensity d p) v) x+ | |
| sphereEigenForcingCoefficient d p mu x*v x=0 := by | |
| obtain ⟨hr,he⟩ := sphere_eigen_rescaled_input d mu hmu f heigen | |
| apply sphere_eigen_matrix_equation d p hrho mu hmu (mu⁻¹ • f) he v hv | |
| simpa only [hr] using ha | |
| theorem sphere_eigen_direct_representative (d : SphereEnergyData) (p : Base) | |
| (hrho : ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x))) | |
| (mu : ℝ) (hmu : mu ≠ 0) | |
| (f : SphereWeightedL2 d) (heigen : sphereL2Resolvent d f=mu • f) : | |
| ∃ v : Yau.Jets.Coord → ℝ, ContDiff ℝ ∞ v ∧ | |
| tsupport v ⊆ Yau.realCenteredCube 4 (1/8) ∧ | |
| v =ᵐ[volume.restrict (Yau.realCenteredCube 4 (1/32))] | |
| (fun x ↦ f (sphereChartCoordMap p x)) ∧ | |
| ∀ x ∈ interior (Yau.realCenteredCube 4 (1/64)), | |
| Yau.coordDiv (realMatrixFlux (sphereChartPrincipalDensity d p) v) x+ | |
| sphereEigenForcingCoefficient d p mu x*v x=0 := by | |
| obtain ⟨hr,he⟩ := sphere_eigen_rescaled_input d mu hmu f heigen | |
| obtain ⟨v,hv,hs,ha⟩ := sphere_eigen_smooth_representative d p hrho mu hmu (mu⁻¹ • f) he | |
| rw [hr] at ha | |
| exact ⟨v,hv,hs,ha,sphere_eigen_direct_matrix_equation d p hrho mu hmu f heigen v hv ha⟩ | |
| end | |
| end Yau.Target | |
| end OAI | |