| import Mathlib | |
| import OAI.Geometry.SmoothYau.Estimates.NormalFamilyInverseExtension | |
| import OAI.Geometry.SmoothYau.Smoothness.FamilyInverseFst | |
| import OAI.Geometry.SmoothYau.Smoothness.UniformJetBoundsFiberComp | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open MvPolynomial | |
| open Set Filter | |
| open scoped ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter Function MvPolynomial | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| open scoped Topology | |
| open Set Filter Manifold Bundle MeasureTheory | |
| open scoped Topology ContDiff ENNReal | |
| open Matrix | |
| open scoped Topology Matrix.Norms.Elementwise | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| namespace YauCounterexamples | |
| section LocalCompTransfer | |
| variable {P E F V : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| theorem compact_fiber_local_comp_bound (ψ : P × E → F) (hψ : ContDiff ℝ ∞ ψ) | |
| {L : Set (P × E)} (hL : IsCompact L) (m : ℕ) : | |
| ∃ C > 0, ∀ q : P, ∀ u : E → V, ∀ f : F → V, ContDiff ℝ ∞ f → | |
| (∀ y ∈ tsupport u, (q,y) ∈ L ∧ u =ᶠ[𝓝 y] (fun z => f (ψ (q,z)))) → | |
| ∀ n : ℝ, 1 ≤ n → ∀ a : E → ℝ, (∀ y, 0 ≤ a y) → | |
| (∀ y ∈ tsupport u, ∀ i ≤ m, | |
| ‖iteratedFDeriv ℝ i f (ψ (q,y))‖ ≤ a y*n^i) → | |
| ∀ y, ∀ k ≤ m, ‖iteratedFDeriv ℝ k u y‖ ≤ C*a y*n^k := by | |
| obtain ⟨C,hC,hb⟩ := compact_fiber_comp_derivative_bound ψ hψ hL m (V := V) | |
| refine ⟨C,hC,?_⟩ | |
| intro q u f hf hu n hn a ha he y k hk | |
| by_cases hy : y ∈ tsupport u | |
| · obtain ⟨hyL,hug⟩ := hu y hy | |
| rw [(hug.iteratedFDeriv ℝ k).eq_of_nhds] | |
| exact hb (q,y) hyL f hf n hn (a y) (ha y) (he y hy) k hk | |
| · have hz := notMem_tsupport_iff_eventuallyEq.mp hy | |
| rw [(hz.iteratedFDeriv ℝ k).eq_of_nhds] | |
| simp only [iteratedFDeriv_zero,Pi.zero_apply,norm_zero] | |
| exact mul_nonneg (mul_nonneg hC.le (ha y)) (pow_nonneg (le_trans zero_le_one hn) _) | |
| end LocalCompTransfer | |
| theorem exists_normal_wave_chart_family (g : SmoothMetric NormalWaveSpace NormalWaveSpace) | |
| {K : Set NormalWaveSpace} (hK : IsCompact K) : | |
| ∃ r > 0, ∃ e : NormalWaveParameter → OpenPartialHomeomorph NormalWaveSpace NormalWaveSpace, | |
| (∀ q, (e q : NormalWaveSpace → NormalWaveSpace) = | |
| normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2)) ∧ | |
| (∀ q, ContDiffOn ℝ ∞ (e q).symm (e q).target) ∧ | |
| (∀ q ∈ metricFrameSet g K, Metric.closedBall (0 : NormalWaveSpace) r ⊆ (e q).source) ∧ | |
| ∃ ψ : NormalWaveParameter × NormalWaveSpace → NormalWaveSpace, ContDiff ℝ ∞ ψ ∧ | |
| ∀ q ∈ metricFrameSet g K, ∀ x ∈ Metric.closedBall (0 : NormalWaveSpace) r, | |
| (e q).symm =ᶠ[𝓝 (e q x)] (fun y => ψ (q,y)) := by | |
| obtain ⟨r,hr,e,he,hs,hi,hR,ψ,hψ,hg⟩ := normal_family_inverse_extension g hK | |
| have he1 : ∀ w, (e w).1 = w.1 := by intro w; rw [he]; rfl | |
| let eq := fiberOpenPartialHomeomorph e he1 | |
| have heq (q : NormalWaveParameter) : (eq q : NormalWaveSpace → NormalWaveSpace) = | |
| normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2) := by | |
| funext x | |
| change (e (q,x)).2 = _ | |
| rw [he] | |
| rfl | |
| refine ⟨r,hr,eq,heq,(fun q => contDiffOn_fiberInverse e he1 q hi),?_,ψ,hψ,?_⟩ | |
| · intro q hq x hx | |
| exact hR q hq x hx | |
| · intro q hq x hx | |
| change (fun y => (e.symm (q,y)).2) =ᶠ[𝓝 (eq q x)] (fun y => ψ (q,y)) | |
| rw [heq] | |
| exact hg q hq x hx | |
| end YauCounterexamples | |
| end | |
| end OAI | |