| import OAI.Geometry.NodalSets.Elliptic.WeightedLift |
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| namespace OAI |
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| namespace Yau.Target |
| open Manifold Matrix Yau.Geometry |
| open scoped ContDiff |
| noncomputable section |
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| lemma circleLift_smooth (f : Base → ℝ) (hf : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ f) : |
| ContMDiff modelWithCorners 𝓘(ℝ,ℝ) ∞ (circleLift f) := |
| hf.comp contMDiff_fst |
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| lemma circleLift_nonzero (f : Base → ℝ) (hf : f ≠ 0) : circleLift f ≠ 0 := by |
| intro h |
| apply hf |
| funext x |
| exact congrFun h (x,1) |
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| lemma independentAmbientMetric_eigenfunction_lift |
| (A : Base → Matrix (Fin 5) (Fin 5) ℝ) (rho : Base → ℝ) |
| (hA : ∀ i j, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ (fun x ↦ A x i j)) |
| (hp : ∀ x, (A x).PosDef) (hr : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ rho) |
| (hrp : ∀ x, 0 < rho x) |
| (hrad : ∀ x : Base, A x *ᵥ (fun i ↦ (x : AmbientBase) i) = (fun i ↦ (x : AmbientBase) i)) |
| (f : Base → ℝ) (hf : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ f) (hf0 : f ≠ 0) |
| (lam : ℝ) |
| (he : ∀ x : Base, -ambientWeightedChartOperator A rho f x (extChartAt (𝓡 4) x x) = lam * f x) : |
| ContMDiff modelWithCorners 𝓘(ℝ,ℝ) ∞ (circleLift f) ∧ circleLift f ≠ 0 ∧ |
| ∀ x : Manifold5, -chartLaplacian (independentAmbientMetric A rho hA hp hr hrp) |
| (extChartAt modelWithCorners x) (circleLift f) (extChartAt modelWithCorners x x) = |
| lam * circleLift f x := by |
| refine ⟨circleLift_smooth f hf,circleLift_nonzero f hf0,fun x ↦ ?_⟩ |
| rw [independentAmbientMetric_weighted_lift A rho hA hp hr hrp hrad f hf x |
| ((extChartAt modelWithCorners x).map_source (mem_extChartAt_source x))] |
| exact he x.1 |
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|
| end |
| end Yau.Target |
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| end OAI |
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