File size: 3,731 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 65 66 67 68 69 70 | import OAI.Geometry.NodalSets.Charts.RoundSphereEnergyData
import OAI.Geometry.NodalSets.Elliptic.IntrinsicOneStep
namespace OAI
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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