| import OAI.Geometry.NodalSets.Charts.SphereWeightedNormalization |
| import OAI.Geometry.NodalSets.Spectral.IntrinsicEigenfunctionLinearity |
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|
| namespace OAI |
|
|
| namespace Yau.Target |
| open Manifold Yau.Geometry |
| open scoped ContDiff |
| noncomputable section |
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|
| lemma intrinsic_eigenfunction_smul (A : IntrinsicTensor) (hA : IntrinsicTensorSmooth A) |
| (hs : ∀ x alpha beta, A x alpha beta = A x beta alpha) |
| (hp : ∀ x alpha, alpha ≠ 0 → 0 < A x alpha alpha) (rho : Base → ℝ) |
| (u : Base → ℝ) (hu : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u) (lam c : ℝ) |
| (he : ∀ p z, -intrinsicWeightedChartOperator A rho u p z = |
| lam*u ((extChartAt (𝓡 4) p).symm z)) : |
| ∀ p z, -intrinsicWeightedChartOperator A rho (c • u) p z = |
| lam*(c • u) ((extChartAt (𝓡 4) p).symm z) := by |
| have hc := intrinsic_eigenfunction_linear_combination A hA hs hp rho u u hu hu lam he he (c-1) |
| have heq : (fun q ↦ (c-1)*u q+u q) = c • u := by |
| funext q |
| simp only [Pi.smul_apply,smul_eq_mul] |
| ring |
| intro p z |
| rw [← heq] |
| exact hc p z |
|
|
| theorem sphere_normalized_orthogonal_eigenfunction (A : IntrinsicTensor) (hA : IntrinsicTensorSmooth A) |
| (hs : ∀ x alpha beta, A x alpha beta = A x beta alpha) |
| (hp : ∀ x alpha, alpha ≠ 0 → 0 < A x alpha alpha) |
| (rho : Base → ℝ) (hr : Continuous rho) (hrp : ∀ x, 0 < rho x) |
| (u v : Base → ℝ) (hu : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u) |
| (hv : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v) (hune : u ≠ 0) (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)) |
| (hnot : ¬ ∃ c : ℝ, v = c • u) : |
| ∃ w : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ w ∧ w ≠ 0 ∧ |
| (∀ p z, -intrinsicWeightedChartOperator A rho w p z = |
| lam*w ((extChartAt (𝓡 4) p).symm z)) ∧ |
| sphereWeightedPairing rho w w = 1 ∧ sphereWeightedPairing rho w u = 0 := by |
| obtain ⟨c,q,hq,hqs,hqo,hqzero⟩ := sphere_weighted_orthogonal_remainder rho u v hr hrp hu hv hune |
| have hqne : q ≠ 0 := fun hz ↦ hnot ⟨c,hqzero.mp hz⟩ |
| have hqeq : q = fun x ↦ (-c)*u x+v x := by |
| rw [hq] |
| funext x |
| simp only [Pi.sub_apply,Pi.smul_apply,smul_eq_mul] |
| ring |
| have hqe : ∀ p z, -intrinsicWeightedChartOperator A rho q p z = |
| lam*q ((extChartAt (𝓡 4) p).symm z) := by |
| rw [hqeq] |
| exact intrinsic_eigenfunction_linear_combination A hA hs hp rho u v hu hv lam heu hev (-c) |
| obtain ⟨s,w,_,hw,hws,hwne,hwn⟩ := sphere_weighted_normalization rho q hr hrp hqs hqne |
| refine ⟨w,hws,hwne,?_,hwn,?_⟩ |
| · rw [hw] |
| exact intrinsic_eigenfunction_smul A hA hs hp rho q hqs lam s hqe |
| · rw [hw,sphereWeightedPairing_smul_left,hqo,mul_zero] |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
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|
