openAI-math-dataset / lean /OAI /Geometry /NodalSets /Persistence /SphereSimpleUniformPersistence.lean
| import Mathlib.Analysis.SpecificLimits.Basic | |
| import OAI.Geometry.NodalSets.Charts.SphereFiniteNormIntrinsicCompactness | |
| import OAI.Geometry.NodalSets.Charts.SphereNormalizedSimpleUniqueness | |
| namespace OAI | |
| namespace Yau.Target | |
| open Manifold MeasureTheory Filter Metric Yau.Geometry | |
| open scoped ContDiff Topology | |
| noncomputable section | |
| theorem sphere_finite_norm_simple_uniform_persistence (d : SphereEnergyData) | |
| (hd : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ d.density) (N : ℕ) | |
| (u : Base → ℝ) (hun : sphereWeightedPairing d.density u u=1) | |
| (hsimple : ∀ v : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v → | |
| (∀ p y, -intrinsicWeightedChartOperator d.tensor d.density v p y = | |
| sphereIndexedEigenvalue d N*v ((extChartAt (𝓡 4) p).symm y)) → ∃ c : ℝ, v=c • u) : | |
| ∃ P : Finset Base, | |
| (∀ x : Base, ∃ p ∈ P, ∃ y ∈ ball (0 : Yau.Jets.Coord) (1/512), sphereChartCoordMap p y=x) ∧ | |
| ∀ eps > 0, ∃ eta > 0, ∀ b : SphereEnergyData, | |
| ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ b.density → | |
| sphereCoefficientDistance P 8 d.tensor d.density b.tensor b.density < eta → | |
| ∀ v : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v → | |
| sphereWeightedPairing b.density v v=1 → | |
| (∀ p y, -intrinsicWeightedChartOperator b.tensor b.density v p y = | |
| sphereIndexedEigenvalue b N*v ((extChartAt (𝓡 4) p).symm y)) → | |
| (∀ x, |v x-u x| < eps) ∨ (∀ x, |v x+u x| < eps) := by | |
| classical | |
| obtain ⟨P,hP,eta₀,heta₀,H⟩ := sphere_finite_norm_indexed_intrinsic_compactness d hd N | |
| refine ⟨P,hP,?_⟩ | |
| intro eps heps | |
| let Good (v : Base → ℝ) : Prop := (∀ x, |v x-u x| < eps) ∨ (∀ x, |v x+u x| < eps) | |
| change ∃ eta > 0, ∀ b : SphereEnergyData, | |
| ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ b.density → | |
| sphereCoefficientDistance P 8 d.tensor d.density b.tensor b.density < eta → | |
| ∀ v : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ v → | |
| sphereWeightedPairing b.density v v=1 → | |
| (∀ p y, -intrinsicWeightedChartOperator b.tensor b.density v p y = | |
| sphereIndexedEigenvalue b N*v ((extChartAt (𝓡 4) p).symm y)) → Good v | |
| by_contra hbad | |
| push Not at hbad | |
| choose b hb hdist v hv hn he hbad using | |
| (fun j : ℕ ↦ hbad (min eta₀ (1/((j:ℝ)+1))) (lt_min heta₀ (by positivity))) | |
| have hcoeff : Tendsto (fun j ↦ sphereCoefficientDistance P 8 d.tensor d.density | |
| (b j).tensor (b j).density) atTop (𝓝 0) := by | |
| apply squeeze_zero _ _ tendsto_one_div_add_atTop_nhds_zero_nat | |
| · intro j | |
| exact sphereCoefficientDistance_nonneg P 8 d.tensor (b j).tensor d.density (b j).density | |
| (fun p _ ↦ intrinsic_coefficient_chart_smooth d.tensor d.smooth d.symm d.pos d.density hd p) | |
| (fun p _ ↦ intrinsic_coefficient_chart_smooth (b j).tensor (b j).smooth (b j).symm (b j).pos | |
| (b j).density (hb j) p) | |
| · intro j | |
| exact (hdist j).le.trans (min_le_right _ _) | |
| obtain ⟨v₀,hv₀,hn₀,_,he₀,nu,hnu,ht,_⟩ := H b hb | |
| (fun j ↦ (hdist j).trans_le (min_le_left _ _)) hcoeff v hv hn he | |
| have hsigned := sphere_normalized_simple_eigenfunction_signed d u hun (sphereIndexedEigenvalue d N) | |
| hsimple v₀ hv₀ hn₀ he₀ | |
| have hc := (Metric.tendstoUniformly_iff.mp ht) eps heps | |
| obtain ⟨j,hj⟩ := hc.exists | |
| apply hbad (nu j) | |
| rcases hsigned with rfl | rfl | |
| · left | |
| intro x | |
| simpa only [Real.dist_eq,abs_sub_comm] using hj x | |
| · right | |
| intro x | |
| simpa only [Pi.neg_apply,Real.dist_eq,abs_sub_comm,sub_neg_eq_add] using hj x | |
| end | |
| end Yau.Target | |
| end OAI | |