| import OAI.Geometry.IsometricImmersion.Estimates.BoundedClassQFloor |
| import OAI.Geometry.IsometricImmersion.Calculus.ShearMetricJets |
| import OAI.Geometry.IsometricImmersion.Darboux.QLowActualBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff Matrix |
|
|
| namespace SmoothLocal.Perturbation |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow |
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|
| theorem exists_bounded_class_sheared_QLowBounds |
| (G R kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G) (hR : 0 ≤ R) |
| (hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) : |
| ∃ MQ : ℝ, 0 ≤ MQ ∧ ∀ (g : MetricField) (z : Coord → ℝ) (S : Set Coord), |
| BoundedAdmissibleHeight g M z → |
| (∀ p ∈ S, inverseShearCoordinates q0 p ∈ modelSquare) → |
| (∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) → |
| (∀ i j k, k ≤ 4 → ∀ p ∈ S, |
| ‖iteratedFDeriv ℝ k (fun r => g r i j) (inverseShearCoordinates q0 p)‖ ≤ G) → |
| (∀ p ∈ S, d ≤ |(g (inverseShearCoordinates q0 p)).det|) → |
| (∀ p ∈ S, gaussianCurvature g (inverseShearCoordinates q0 p) ≤ -kappa / 2) → |
| (∀ p ∈ S, |hessianQuotient g z (inverseShearCoordinates q0 p) - q0| ≤ |
| 1 / (10 * boundedClassWidth kappa M)) → |
| QLowBounds (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) S |
| (shearedQPositiveFloor G (M : ℝ) d q0 (M : ℝ)) MQ (boundedClassWidth kappa M / 4) := by |
| let GS := shearedMetricCoordinateBudget G q0 4 |
| let ZS := (2 * (1 + |q0|))^3 * (M : ℝ) |
| let c := (boundedClassSpeed kappa M)^2 / (2 * (M : ℝ)) |
| have hGS : 0 ≤ GS := shearedMetricCoordinateBudget_nonneg q0 4 hG |
| have hc : 0 < c := boundedClassShearHxxFloor_pos hkappa hM |
| obtain ⟨MQ, hMQ, hlow⟩ := exists_actual_QLowBounds GS R ZS hGS hR hd hc |
| refine ⟨MQ, hMQ, ?_⟩ |
| intro g z S hclass hSp hcoords hgB hdet hK hq |
| have hclass0 := hclass |
| obtain ⟨_, hadm, hjet, _, _⟩ := hclass |
| obtain ⟨U, hU, hSU, hg, hz, _, _, _, _⟩ := hadm |
| let T : Set Coord := inverseShearCoordinates q0 '' S |
| let V : Set Coord := inverseShearCoordinates q0 ⁻¹' U |
| have hTU : T ⊆ U := by |
| rintro p ⟨r, hr, rfl⟩ |
| exact hSU (hSp r hr) |
| have hSV : S ⊆ V := fun p hp => hSU (hSp p hp) |
| have hgCoord : ∀ i j, CoordinateBound (fun p => g p i j) T 4 G := by |
| intro i j ds hds p hp |
| obtain ⟨r, hr, rfl⟩ := hp |
| exact (norm_iteratedCoordPartial_le_jet (hg.1 i j) hU ds (hSU (hSp r hr))).trans |
| (hgB i j ds.length hds r hr) |
| have hgShear := metric_coordinate_bound_inverse_shear hg hU hTU hG hgCoord q0 |
| have hgS : ∀ i j, CoordinateBound (fun p => metricInShearCoordinates g q0 p i j) S 4 GS := by |
| intro i j ds hds p hp |
| exact hgShear i j ds hds p ⟨p, hp, rfl⟩ |
| have hzCoord : CoordinateBound z modelSquare 3 (M : ℝ) := by |
| intro ds hds p hp |
| exact (norm_iteratedCoordPartial_le_jet hz hU ds (hSU hp)).trans |
| (hjet ds.length (by omega) p hp) |
| have hzShear := SmoothLocal.Pulse.CoordinateBound.inverse_shear hzCoord hz hU hSU (Nat.cast_nonneg M) q0 |
| have hzS : CoordinateBound (heightInShearCoordinates z q0) S 3 ZS := by |
| intro ds hds p hp |
| exact hzShear ds hds p (hSp p hp) |
| have hgSmooth := metricInShearCoordinates_smoothPositive hg q0 |
| have hVS : IsOpen V := hU.preimage (inverseShearCoordinates_contDiff q0).continuous |
| have hzSmooth := heightInShearCoordinates_contDiffOn hz q0 |
| obtain ⟨_, hprincipal⟩ := bounded_class_Q_principal_budgets hclass0 hkappa hG hd hSp |
| (fun i j k hk => hgB i j k (by omega)) hdet hK hq |
| exact hlow (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) V S |
| hgSmooth hVS hzSmooth hSV hcoords hgS hzS |
| (fun p hp => by simpa only [metricDet_in_shear_coordinates] using hdet p hp) |
| (fun p hp => (hprincipal p hp).1) |
| (shearedQPositiveFloor G (M : ℝ) d q0 (M : ℝ)) (boundedClassWidth kappa M / 4) |
| (fun p hp => (hprincipal p hp).2.1) (fun p hp => (hprincipal p hp).2.2) |
|
|
| end SmoothLocal.Perturbation |
|
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| end |
|
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| end OAI |
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|
