| import OAI.Geometry.NodalSets.Charts.RoundSphereEnergyData |
| import OAI.Geometry.NodalSets.Elliptic.IntrinsicOneStep |
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|
| namespace OAI |
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|
| namespace Yau.Target |
| open Manifold Yau.Geometry Set Metric |
| open scoped ContDiff RealInnerProductSpace ENNReal |
| noncomputable section |
|
|
| structure SphereComparisonPair (r delta : ℝ) where |
| energy : SphereEnergyData |
| density_smooth : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ energy.density |
| comparison : ∀ y ∈ closedBall (0 : BaseModel) r, |
| dist ((intrinsicChartCoefficient energy.tensor energy.density seedPoint y, |
| fderiv ℝ (intrinsicChartCoefficient energy.tensor energy.density seedPoint) y) : CoefficientFirstJet BaseModel) |
| (roundCoefficientJet y) < delta |
| support : Set Base |
| support_compact : IsCompact support |
| support_patch : support ⊆ seedSpherePatch r |
| exterior_tensor : ∀ x ∉ support, ∀ v w : AmbientBase, ⟪(x:AmbientBase),v⟫=0 → ⟪(x:AmbientBase),w⟫=0 → |
| energy.tensor x (sphereCovectorRestriction x v) (sphereCovectorRestriction x w)=⟪v,w⟫ |
| exterior_density : ∀ x ∉ support, energy.density x=1 |
|
|
| structure SphereNodalStage (r delta : ℝ) (k : ℕ) extends SphereComparisonPair r delta where |
| frequency : ℕ |
| frequency_pos : 0 < frequency |
| charts : Finset Base |
| radius : ℝ |
| radius_pos : 0 < radius |
| persistence : ∀ (b : SphereEnergyData) (hb : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ b.density), |
| sphereCoefficientDistance charts 8 energy.tensor energy.density b.tensor b.density < radius → |
| ∃ (mu : ℝ) (v : Base → ℝ), mu ∈ Icc (seedEigenvalue frequency/2) (2*seedEigenvalue frequency) ∧ |
| 0 < mu ∧ ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v ∧ v ≠ 0 ∧ |
| (∀ p y, -intrinsicWeightedChartOperator b.tensor b.density v p y = |
| mu*v ((extChartAt (𝓡 4) p).symm y)) ∧ |
| ContMDiff modelWithCorners 𝓘(ℝ,ℝ) ∞ (circleLift v) ∧ circleLift v ≠ 0 ∧ |
| (∀ x : Manifold5, -chartLaplacian |
| (intrinsicWeightedMetric b.tensor b.smooth b.symm b.pos b.density hb b.positive) |
| (extChartAt modelWithCorners x) (circleLift v) (extChartAt modelWithCorners x x) = |
| mu*circleLift v x) ∧ |
| ENNReal.ofReal ((k:ℝ)+1) < nodalMeasure |
| (intrinsicWeightedMetric b.tensor b.smooth b.symm b.pos b.density hb b.positive) |
| (circleLift v) / ENNReal.ofReal (Real.sqrt mu) |
|
|
| def SphereStageExtension (r delta : ℝ) : Prop := |
| ∀ (d : SphereComparisonPair r delta) (k : ℕ) (P : Finset Base) (J : ℕ) (eps : ℝ), 0 < eps → |
| ∀ n₀ : ℕ, ∃ s : SphereNodalStage r delta k, n₀ ≤ s.frequency ∧ d.support ⊆ s.support ∧ |
| sphereCoefficientDistance P J d.energy.tensor d.energy.density s.energy.tensor s.energy.density < eps |
|
|
| theorem sphere_iteration_interface : |
| ∃ r delta : ℝ, 0 < r ∧ 0 < delta ∧ |
| (∃ d : SphereComparisonPair r delta, d.energy=roundSphereEnergyData ∧ d.support=∅) ∧ |
| SphereStageExtension r delta := by |
| obtain ⟨r,a,delta,hr,ha,hdelta,hcube,hbranch,H⟩ := intrinsic_one_step |
| refine ⟨r,delta,hr,hdelta,?_,?_⟩ |
| · exact ⟨⟨roundSphereEnergyData,roundSphereEnergyData_density_smooth, |
| roundSphereEnergyData_comparison r hdelta,∅,isCompact_empty,empty_subset _, |
| roundSphereEnergyData_exterior.1,roundSphereEnergyData_exterior.2⟩,rfl,rfl⟩ |
| · intro d k P J eps heps n₀ |
| obtain ⟨n,hn,hn0,b,hb,hclose,hcomp,K,hK,hsub,hKP,hA,hrho,Q,eta,heta,Heta⟩ := |
| H d.energy d.density_smooth d.comparison d.support d.support_compact d.support_patch |
| d.exterior_tensor d.exterior_density ((k:ℝ)+1) (by positivity) P J eps heps n₀ |
| exact ⟨⟨⟨b,hb,hcomp,K,hK,hKP,hA,hrho⟩,n,hn0,Q,eta,heta,Heta⟩,hn,hsub,hclose⟩ |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
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|
