File size: 2,875 Bytes
8fadf68 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Geometry.NodalSets.Charts.SphereWeightedNormalization
import OAI.Geometry.NodalSets.Spectral.IntrinsicEigenfunctionLinearity
namespace OAI
namespace Yau.Target
open Manifold Yau.Geometry
open scoped ContDiff
noncomputable section
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
|