File size: 3,485 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 | import OAI.Geometry.IsometricImmersion.Immersions.BoundedHeightClasses
import OAI.Geometry.IsometricImmersion.Comparison.PulseClosureApproximation
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse
theorem boundedHeightClass_closure_exists_approximation
{g0 : MetricField} {U : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 U) (M : ℕ) (q0 : ℚ)
{theta : metricPatchSet g0 kappa}
(hclosure : theta ∈ closure (boundedHeightClass g0 kappa M q0))
(n : ℕ) {epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z ∧
|hessianQuotient (perturbedMetric g0 eta.val) z 0 - (q0 : ℝ)| <
1 / (100 * boundedClassWidth kappa M) ∧
SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧
(∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x -
perturbedMetric g0 theta.val x) p‖ ≤ epsilon) ∧
(∀ i j : Fin 2, ∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j -
perturbedMetric g0 theta.val x i j) p‖ ≤ epsilon) := by
obtain ⟨eta, heta, hpositive, _, hT, hC⟩ :=
relative_closure_exists_metric_approximation hg0 hclosure n hepsilon
obtain ⟨z, hz, hlabel⟩ := heta
exact ⟨eta, z, hz, hlabel, hpositive, hT, hC⟩
theorem exists_N_before_delta_eventually_bounded_height_approximation
{g0 : MetricField} {U : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(M : ℕ) (q0 : ℚ)
{O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O)
{etaStar : metricPatchSet g0 kappa} (hStar : etaStar ∈ O)
(hOclosure : O ⊆ closure (boundedHeightClass g0 kappa M q0))
(a : ℝ) (ha : 0 < a) (hax : a ≤ 1 / 10) (hq : |(q0 : ℝ)| * a < 1 / 10) :
∃ N : ℕ, 10 < N ∧ ∀ delta : ℝ, 0 < delta →
∀ᶠ tau : ℕ in atTop, 0 < tau ∧
∃ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z ∧
|hessianQuotient (perturbedMetric g0 eta.val) z 0 - (q0 : ℝ)| <
1 / (100 * boundedClassWidth kappa M) ∧
SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧
(∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x -
testMetric (perturbedMetric g0 etaStar.val) (q0 : ℝ) a N delta (tau : ℝ) x) p‖ ≤
metricApproximationAccuracy tau) ∧
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j -
testMetric (perturbedMetric g0 etaStar.val) (q0 : ℝ) a N delta (tau : ℝ) x i j) p‖ ≤
metricApproximationAccuracy tau) := by
obtain ⟨N, hN, happrox⟩ := exists_N_before_delta_eventually_class_approximation
hg0 hU hSU hO hStar hOclosure (q0 : ℝ) a ha hax hq
refine ⟨N, hN, ?_⟩
intro delta hdelta
filter_upwards [happrox delta hdelta] with tau htau
obtain ⟨eta, ⟨z, hz, hlabel⟩, hpositive, _, hT, hC⟩ := htau.2
exact ⟨htau.1, eta, z, hz, hlabel, hpositive, hT, hC⟩
end SmoothLocal.Perturbation
end
end OAI
|