File size: 3,989 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 | 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
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
end
end OAI
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