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59d1c80 | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Geometry.IsometricImmersion.Estimates.BoundedClassQLow
import OAI.Geometry.IsometricImmersion.Pulses.PulseFlowDistance
import OAI.Geometry.IsometricImmersion.Calculus.ClosedSquareJetBounds
namespace OAI
noncomputable section
open Set
open scoped ContDiff Topology Matrix
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow
def boundedClassQuotientBudget (G d : ℝ) (M : ℕ) : ℝ :=
heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ))
def boundedClassSlabRadius (G d kappa q0 : ℝ) (M : ℕ) : ℝ :=
1 / (100 * boundedClassWidth kappa M * (1 + boundedClassQuotientBudget G d M) *
(1 + (1 + |q0|) * boundedClassWidth kappa M))
theorem boundedClassQuotientBudget_nonneg {G d : ℝ} {M : ℕ}
(hG : 0 ≤ G) (hd : 0 < d) (hM : 0 < M) : 0 ≤ boundedClassQuotientBudget G d M :=
heightQuotientJetBound_nonneg hG (Nat.cast_nonneg M) hd
(one_div_pos.mpr (Nat.cast_pos.mpr hM))
theorem boundedClassSlabRadius_pos {G d : ℝ} (kappa q0 : ℝ) {M : ℕ}
(hG : 0 ≤ G) (hd : 0 < d) (hM : 0 < M) : 0 < boundedClassSlabRadius G d kappa q0 M := by
have hL := boundedClassWidth_pos kappa M
have hQ := boundedClassQuotientBudget_nonneg hG hd hM
unfold boundedClassSlabRadius
positivity
theorem bounded_class_quotient_change_from_origin
{g : MetricField} {z : Coord → ℝ} {M : ℕ} (h : BoundedAdmissibleHeight g M z)
{G d : ℝ} (hG : 0 ≤ G) (hd : 0 < d)
(hgB : ∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun r => g r i j) p‖ ≤ G)
(hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|)
{p : Coord} (hp : p ∈ modelSquare) :
|hessianQuotient g z p - hessianQuotient g z 0| ≤
boundedClassQuotientBudget G d M * (|p 0| + |p 1|) := by
obtain ⟨hM, hadm, hjet, hHb, _⟩ := h
obtain ⟨U, hU, hSU, hg, hz, _, _, _, _⟩ := hadm
have hgCoord : ∀ i j, CoordinateBound (fun p => g p i j) modelSquare 4 G := by
intro i j ds hds p hp
exact (norm_iteratedCoordPartial_le_jet (hg.1 i j) hU ds (hSU hp)).trans
(hgB i j ds.length hds p hp)
have hzCoord : CoordinateBound z modelSquare 5 (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)
obtain ⟨V, hV, hSV, _, _, hqV, hqB⟩ :=
exists_open_quotient_domain_with_closed_low_bounds hg hU hSU hz hG (Nat.cast_nonneg M) hd
(one_div_pos.mpr (Nat.cast_pos.mpr hM)) hgCoord hzCoord hdet (fun p hp => (hHb p hp).1)
apply quotient_change_from_origin hqV hV hSV (boundedClassQuotientBudget_nonneg hG hd hM) ?_ hp
intro p hp i
exact hqB [i] (by norm_num) p hp
theorem inverseShear_coordinate_sum_le {q0 L r : ℝ} {p : Coord}
(_hr : 0 ≤ r) (hx : |p 0| ≤ L * r) (hy : |p 1| ≤ r) :
|inverseShearCoordinates q0 p 0| + |inverseShearCoordinates q0 p 1| ≤
r * (1 + (1 + |q0|) * L) := by
change |p 0| + |p 1 - q0 * p 0| ≤ _
calc
_ ≤ |p 0| + (|p 1| + |q0 * p 0|) := add_le_add le_rfl (abs_sub _ _)
_ ≤ L * r + (r + |q0| * (L * r)) := by
rw [abs_mul]
exact add_le_add hx (add_le_add hy (mul_le_mul_of_nonneg_left hx (abs_nonneg _)))
_ = _ := by ring
theorem bounded_class_slab_quotient_small
{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)
(hrsmall : r ≤ boundedClassSlabRadius G d kappa q0 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
let D := 1 + (1 + |q0|) * L
have hL : 0 < L := boundedClassWidth_pos kappa M
have hQ : 0 ≤ Q := boundedClassQuotientBudget_nonneg hG hd h.1
have hD : 0 < D := by dsimp [D]; positivity
have hbudget : Q * (r * D) ≤ 1 / (100 * L) := by
calc
_ ≤ (1 + Q) * (r * D) := mul_le_mul_of_nonneg_right (by linarith) (mul_nonneg hr hD.le)
_ ≤ (1 + Q) * (boundedClassSlabRadius G d kappa q0 M * D) :=
mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hrsmall hD.le) (by positivity)
_ = _ := by
change (1 + Q) * (1 / (100 * L * (1 + Q) * D) * D) = 1 / (100 * L)
field_simp [hL.ne', hD.ne', ne_of_gt (show 0 < 1 + Q by positivity)]
have hchange := bounded_class_quotient_change_from_origin h hG hd hgB hdet hp
have hdist := inverseShear_coordinate_sum_le (q0 := q0) hr hx hy
have hchange' : |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| ≤
1 / (100 * L) := hchange.trans ((mul_le_mul_of_nonneg_left hdist hQ).trans hbudget)
calc
_ ≤ |hessianQuotient g z (inverseShearCoordinates q0 p) - hessianQuotient g z 0| +
|hessianQuotient g z 0 - q0| := abs_sub_le _ _ _
_ ≤ 1 / (100 * L) + 1 / (100 * L) := add_le_add hchange' hcenter
_ ≤ 1 / (10 * L) := by
have hInv : 0 < 1 / L := one_div_pos.mpr hL
calc
_ = (1 / 50 : ℝ) * (1 / L) := by
field_simp [hL.ne']
ring
_ ≤ (1 / 10 : ℝ) * (1 / L) :=
mul_le_mul_of_nonneg_right (by norm_num) hInv.le
_ = _ := by field_simp [hL.ne']
theorem exists_bounded_class_slab_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) (r : ℝ),
BoundedAdmissibleHeight g M z → 0 ≤ r → r ≤ boundedClassSlabRadius G d kappa q0 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 hclass hG hd hr hrsmall hgB hdet hcenter
(hSp p hp) (haxes p hp).1 (haxes p hp).2)
end SmoothLocal.Perturbation
end
end OAI
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