File size: 4,123 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 | 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
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