| import OAI.Geometry.NodalSets.Elliptic.IntrinsicRealCoordinates |
| import OAI.Geometry.NodalSets.Elliptic.RealEllipticLinearity |
|
|
| namespace OAI |
|
|
| namespace Yau.Target |
| open Manifold Yau.Geometry |
| open scoped ContDiff |
| noncomputable section |
|
|
| lemma intrinsicWeightedChartOperator_linear_combination (A : IntrinsicTensor) |
| (hA : IntrinsicTensorSmooth A) (hs : ∀ p v w, A p v w = A p w v) |
| (hp : ∀ p v, v ≠ 0 → 0 < A p v v) (rho : Base → ℝ) |
| (u v : Base → ℝ) (hu : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u) (hv : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v) |
| (c : ℝ) (p : Base) (z : BaseModel) : |
| intrinsicWeightedChartOperator A rho (fun q ↦ c*u q+v q) p z = |
| c*intrinsicWeightedChartOperator A rho u p z+intrinsicWeightedChartOperator A rho v p z := by |
| obtain ⟨x,rfl⟩ := seedCoordEquiv.surjective z |
| rw [intrinsicRealOperator_pullback,intrinsicRealOperator_pullback,intrinsicRealOperator_pullback] |
| change (rho (sphereChartCoordMap p x))⁻¹ * |
| realWeightedElliptic roundCoordDensity (intrinsicRealPrincipal A p) |
| (fun y ↦ c*(u ∘ sphereChartCoordMap p) y+(v ∘ sphereChartCoordMap p) y) x = |
| c*((rho (sphereChartCoordMap p x))⁻¹ * realWeightedElliptic roundCoordDensity |
| (intrinsicRealPrincipal A p) (u ∘ sphereChartCoordMap p) x)+ |
| (rho (sphereChartCoordMap p x))⁻¹ * realWeightedElliptic roundCoordDensity |
| (intrinsicRealPrincipal A p) (v ∘ sphereChartCoordMap p) x |
| rw [realWeightedElliptic_linear_combination _ _ _ _ roundCoordDensity_smooth |
| (intrinsicRealPrincipal_smooth A hA hs hp p) (spherePullback_smooth u hu p) (spherePullback_smooth v hv p)] |
| ring |
|
|
| theorem intrinsic_eigenfunction_linear_combination (A : IntrinsicTensor) |
| (hA : IntrinsicTensorSmooth A) (hs : ∀ p v w, A p v w = A p w v) |
| (hp : ∀ p v, v ≠ 0 → 0 < A p v v) (rho : Base → ℝ) |
| (u v : Base → ℝ) (hu : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u) (hv : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v) |
| (lam : ℝ) |
| (heu : ∀ p z, -intrinsicWeightedChartOperator A rho u p z = lam*u ((extChartAt (𝓡 4) p).symm z)) |
| (hev : ∀ p z, -intrinsicWeightedChartOperator A rho v p z = lam*v ((extChartAt (𝓡 4) p).symm z)) |
| (c : ℝ) : ∀ p z, -intrinsicWeightedChartOperator A rho (fun q ↦ c*u q+v q) p z = |
| lam*(c*u ((extChartAt (𝓡 4) p).symm z)+v ((extChartAt (𝓡 4) p).symm z)) := by |
| intro p z |
| rw [intrinsicWeightedChartOperator_linear_combination A hA hs hp rho u v hu hv] |
| linear_combination c*heu p z+hev p z |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
|
|
