File size: 5,573 Bytes
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 | import OAI.Geometry.IsometricImmersion.Caps.BoundedClassCapRadius
import OAI.Geometry.IsometricImmersion.Metrics.MetricPatchOpen
namespace OAI
noncomputable section
open Set
open scoped ContDiff Matrix
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow
open SmoothLocal.ODE SmoothLocal.Hyperbolic
theorem inverseShear_slab_mem_centralBox {L r q0 : ℝ} {p : Coord}
(hL : 1 ≤ L) (hr : 0 ≤ r) (hLr : L * r ≤ 1 / 20) (hq0 : |q0| ≤ 1 / 20)
(hx : |p 0| ≤ L * r) (hy : |p 1| ≤ r) :
inverseShearCoordinates q0 p ∈ centralBox := by
have hrL : r ≤ L * r := by nlinarith
have hrsmall : r ≤ 1 / 20 := hrL.trans hLr
have hxsmall : |p 0| ≤ 1 / 20 := hx.trans hLr
have hprod : |q0 * p 0| ≤ (1 / 20 : ℝ) * (1 / 20) := by
rw [abs_mul]
exact mul_le_mul hq0 hxsmall (abs_nonneg _) (by norm_num)
have hysmall : |p 1 - q0 * p 0| ≤ (1 / 20 : ℝ) + (1 / 20) * (1 / 20) :=
(abs_sub _ _).trans (add_le_add (hy.trans hrsmall) hprod)
change (fun _ => -(1 / 5 : ℝ)) ≤ inverseShearCoordinates q0 p ∧
inverseShearCoordinates q0 p ≤ (fun _ => (1 / 5 : ℝ))
constructor
· intro i
fin_cases i
· change -(1 / 5 : ℝ) ≤ p 0
linarith [(abs_le.mp hxsmall).1]
· change -(1 / 5 : ℝ) ≤ p 1 - q0 * p 0
linarith [(abs_le.mp hysmall).1]
· intro i
fin_cases i
· change p 0 ≤ (1 / 5 : ℝ)
linarith [(abs_le.mp hxsmall).2]
· change p 1 - q0 * p 0 ≤ (1 / 5 : ℝ)
linarith [(abs_le.mp hysmall).2]
theorem shrinking_class_slab_axis_bounds {kappa r b : ℝ} {M : ℕ} {p : Coord}
(hr : 0 ≤ r) (hb : b ≤ 0)
(hp : p ∈ shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r)
(-r) b (boundedClassWidth kappa M / 4)) :
|p 0| ≤ boundedClassWidth kappa M * r ∧ |p 1| ≤ r := by
have ht := hp.1
have hx := hp.2
have hspeed : 0 ≤ boundedClassWidth kappa M / 4 :=
div_nonneg (boundedClassWidth_pos kappa M).le (by norm_num)
have hmove : 0 ≤ (boundedClassWidth kappa M / 4) * (p 1 - (-r)) :=
mul_nonneg hspeed (sub_nonneg.mpr ht.1)
unfold inwardLeft inwardRight at hx
exact ⟨abs_le.mpr ⟨by linarith [hx.1], by linarith [hx.2]⟩,
abs_le.mpr ⟨ht.1, by linarith [ht.2]⟩⟩
theorem actual_patch_curvature_on_class_slab {g0 : MetricField} {kappa q0 r b : ℝ} {M : ℕ}
(eta : metricPatchSet g0 kappa) (hr : 0 ≤ r) (hb : b ≤ 0)
(hLr : boundedClassWidth kappa M * r ≤ 1 / 20) (hq0 : |q0| ≤ 1 / 20)
{p : Coord}
(hp : p ∈ shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r)
(-r) b (boundedClassWidth kappa M / 4)) :
inverseShearCoordinates q0 p ∈ centralBox ∧
gaussianCurvature (perturbedMetric g0 eta.val) (inverseShearCoordinates q0 p) < -kappa / 2 := by
obtain ⟨hx, hy⟩ := shrinking_class_slab_axis_bounds hr hb hp
have hc := inverseShear_slab_mem_centralBox
((by norm_num : (1 : ℝ) ≤ 100).trans (boundedClassWidth_ge_hundred kappa M)) hr hLr hq0 hx hy
exact ⟨hc, eta.property.2 _ hc⟩
theorem exists_bounded_patch_actual_slab_QLowBounds
(G kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G)
(hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) (hq0 : |q0| ≤ 1 / 20) :
∃ MQ : ℝ, 0 ≤ MQ ∧ ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa)
(z : Coord → ℝ) (r b : ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z → 0 ≤ r → b ≤ 0 →
boundedClassWidth kappa M * r ≤ 1 / 20 →
heightQuotientJetBound G (M : ℝ) d (1 / (M : ℝ)) *
(r + 107 * (boundedClassWidth kappa M * r) / 100) ≤
9 / (100 * boundedClassWidth kappa M) →
(∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun a => perturbedMetric g0 eta.val a i j) p‖ ≤ G) →
(∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) →
|hessianQuotient (perturbedMetric g0 eta.val) z 0 - q0| ≤
1 / (100 * boundedClassWidth kappa M) →
QLowBounds (metricInShearCoordinates (perturbedMetric g0 eta.val) q0)
(heightInShearCoordinates z q0)
(shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r)
(-r) b (boundedClassWidth kappa M / 4))
(shearedQPositiveFloor G (M : ℝ) d q0 (M : ℝ)) MQ (boundedClassWidth kappa M / 4) := by
obtain ⟨MQ, hMQ, hlow⟩ := exists_bounded_class_cap_radius_QLowBounds
G 1 kappa d q0 M hG (by norm_num) hkappa hd hM hq0
refine ⟨MQ, hMQ, ?_⟩
intro g0 eta z r b hclass hr hb hLr hrsmall hgB hdet hcenter
let S := shrinkingSlab (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r)
(-r) b (boundedClassWidth kappa M / 4)
have haxes (p : Coord) (hp : p ∈ S) := shrinking_class_slab_axis_bounds hr hb hp
have hcentral (p : Coord) (hp : p ∈ S) := actual_patch_curvature_on_class_slab eta hr hb hLr hq0 hp
have hrsmall' : r ≤ 1 / 20 := by
have hL : 1 ≤ boundedClassWidth kappa M :=
(by norm_num : (1 : ℝ) ≤ 100).trans (boundedClassWidth_ge_hundred kappa M)
have hh : r ≤ boundedClassWidth kappa M * r := by nlinarith
exact hh.trans hLr
exact hlow (perturbedMetric g0 eta.val) z S r hclass hr hrsmall
(fun p hp => centralBox_subset_modelSquare (hcentral p hp).1)
(fun p hp => ⟨by linarith [(haxes p hp).1], by linarith [(haxes p hp).2]⟩)
haxes hgB hdet (fun p hp => (hcentral p hp).2.le) hcenter
end SmoothLocal.Perturbation
end
end OAI
|