| import OAI.Geometry.IsometricImmersion.Coordinates.BoundedClassSlabSmallness |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff Matrix |
|
|
| namespace SmoothLocal.Perturbation |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow |
|
|
| theorem bounded_class_slab_quotient_small_of_cap_radius |
| {g : MetricField} {z : Coord → ℝ} {M : ℕ} (h : BoundedAdmissibleHeight g M z) |
| {G d kappa q0 r : ℝ} (hG : 0 ≤ G) (hd : 0 < d) (hr : 0 ≤ r) |
| (hq0 : |q0| ≤ 1 / 20) |
| (hrsmall : heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ)) * |
| (r + 107 * (boundedClassWidth kappa M * r) / 100) ≤ 9 / (100 * boundedClassWidth kappa M)) |
| (hgB : ∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, |
| ‖iteratedFDeriv ℝ k (fun a => g a i j) p‖ ≤ G) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) |
| (hcenter : |hessianQuotient g z 0 - q0| ≤ 1 / (100 * boundedClassWidth kappa M)) |
| {p : Coord} (hp : inverseShearCoordinates q0 p ∈ modelSquare) |
| (hx : |p 0| ≤ boundedClassWidth kappa M * r) (hy : |p 1| ≤ r) : |
| |hessianQuotient g z (inverseShearCoordinates q0 p) - q0| ≤ |
| 1 / (10 * boundedClassWidth kappa M) := by |
| let L := boundedClassWidth kappa M |
| let Q := boundedClassQuotientBudget G d M |
| have hL : 0 < L := boundedClassWidth_pos kappa M |
| have hQ : 0 ≤ Q := boundedClassQuotientBudget_nonneg hG hd h.1 |
| have hLr : 0 ≤ L * r := mul_nonneg hL.le hr |
| have hdist := inverseShear_coordinate_sum_le (q0 := q0) hr hx hy |
| have hdist' : |inverseShearCoordinates q0 p 0| + |inverseShearCoordinates q0 p 1| ≤ |
| r + 107 * (L * r) / 100 := by |
| apply hdist.trans |
| have hh := mul_le_mul_of_nonneg_right hq0 hLr |
| nlinarith |
| have hchange := bounded_class_quotient_change_from_origin h hG hd hgB hdet hp |
| have hchange' : |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| ≤ |
| 9 / (100 * L) := hchange.trans ((mul_le_mul_of_nonneg_left hdist' hQ).trans hrsmall) |
| calc |
| _ ≤ |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| + |
| |hessianQuotient g z 0 - q0| := abs_sub_le _ _ _ |
| _ ≤ 9 / (100 * L) + 1 / (100 * L) := add_le_add hchange' hcenter |
| _ = _ := by |
| change 9 / (100 * L) + 1 / (100 * L) = 1 / (10 * L) |
| field_simp [hL.ne'] |
| ring |
|
|
| theorem exists_bounded_class_cap_radius_QLowBounds |
| (G R kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G) (hR : 0 ≤ R) |
| (hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) (hq0 : |q0| ≤ 1 / 20) : |
| ∃ MQ : ℝ, 0 ≤ MQ ∧ ∀ (g : MetricField) (z : Coord → ℝ) (S : Set Coord) (r : ℝ), |
| BoundedAdmissibleHeight g M z → 0 ≤ r → |
| heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ)) * |
| (r + 107 * (boundedClassWidth kappa M * r) / 100) ≤ |
| 9 / (100 * boundedClassWidth kappa M) → |
| (∀ p ∈ S, inverseShearCoordinates q0 p ∈ modelSquare) → |
| (∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) → |
| (∀ p ∈ S, |p 0| ≤ boundedClassWidth kappa M * r ∧ |p 1| ≤ r) → |
| (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, |
| ‖iteratedFDeriv ℝ k (fun a => g a i j) p‖ ≤ G) → |
| (∀ p ∈ modelSquare, d ≤ |(g p).det|) → |
| (∀ p ∈ S, gaussianCurvature g (inverseShearCoordinates q0 p) ≤ -kappa / 2) → |
| |hessianQuotient g z 0 - q0| ≤ 1 / (100 * 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 |
| obtain ⟨MQ, hMQ, hlow⟩ := exists_bounded_class_sheared_QLowBounds G R kappa d q0 M hG hR hkappa hd hM |
| refine ⟨MQ, hMQ, ?_⟩ |
| intro g z S r hclass hr hrsmall hSp hcoords haxes hgB hdet hK hcenter |
| exact hlow g z S hclass hSp hcoords |
| (fun i j k hk p hp => hgB i j k hk _ (hSp p hp)) |
| (fun p hp => hdet _ (hSp p hp)) hK |
| (fun p hp => bounded_class_slab_quotient_small_of_cap_radius hclass hG hd hr hq0 hrsmall |
| hgB hdet hcenter (hSp p hp) (haxes p hp).1 (haxes p hp).2) |
|
|
| end SmoothLocal.Perturbation |
|
|
| end |
|
|
| end OAI |
|
|
