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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 | import OAI.Geometry.NodalSets.Elliptic.WeightedLift
namespace OAI
namespace Yau.Target
open Manifold Matrix Yau.Geometry
open scoped ContDiff
noncomputable section
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
lemma circleLift_nonzero (f : Base → ℝ) (hf : f ≠ 0) : circleLift f ≠ 0 := by
intro h
apply hf
funext x
exact congrFun h (x,1)
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
end
end Yau.Target
end OAI
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