| import OAI.Geometry.IsometricImmersion.Caps.BoundedPatchCauchy |
| import OAI.Geometry.IsometricImmersion.Calculus.ShearedMetricFiniteJets |
| import OAI.Geometry.IsometricImmersion.Estimates.FixedRadiusCutBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology Matrix NNReal |
|
|
| namespace SmoothLocal.Perturbation |
| open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow |
| open SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor |
|
|
| theorem exists_actual_off_pulse_cauchy_bounds_at_radius |
| {gStar : MetricField} {V : Set Coord} |
| (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) |
| {G d kappa q0 r : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d) |
| (hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20) |
| (hr : 0 < r) (hrhalf : r < 1/2) (hLr : boundedClassWidth kappa M*r ≤ 1/20) |
| (hrsmall : heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))* |
| (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M)) |
| (K : ℕ) : |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ N : ℕ, ∀ delta : ℝ, 0 < delta → |
| ∀ᶠ tau : ℕ in atTop, |
| ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U W : Set Coord) |
| (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ), |
| SmoothPositiveOn (perturbedMetric g0 eta.val) U → IsOpen U → |
| CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z → |
| CapInductionFlow (perturbedMetric g0 eta.val) U G (M : ℝ) d (1/(M : ℝ)) (1/(M : ℝ)) kappa z Y W → |
| BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z → |
| (∀ 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) → |
| (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare, |
| ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j- |
| testMetric gStar q0 (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) q i j) p‖ ≤ |
| metricApproximationAccuracy tau) → |
| ∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ n ≤ K, |
| ‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0) |
| (-delta/(tau : ℝ))) x‖ ≤ B ∧ |
| ‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0) |
| (-delta/(tau : ℝ))) x‖ ≤ B := by |
| let L := boundedClassWidth kappa M |
| have hL : 0 < L := boundedClassWidth_pos kappa M |
| have hMR : (0 : ℝ) < M := Nat.cast_pos.mpr hM |
| obtain ⟨A0,hA0,hcutK⟩ := exists_actual_finite_sheared_lower_cut_bound_at_radius |
| (G := G) (Z := (M : ℝ)) (d := d) (c := 1/(M : ℝ)) (e0 := 1/(M : ℝ)) |
| (kappa := kappa) (q0 := q0) (L := L) hgStar hV hSV hG (Nat.cast_nonneg M) hd |
| (one_div_pos.mpr hMR) hL hq0 hr hrhalf hLr hrsmall (max 8 (K+2)) |
| let A : ℝ≥0 := ⟨A0,hA0⟩ |
| obtain ⟨Cmetric,hCmetric,hmetricK⟩ := exists_uniform_off_pulse_sheared_metric_bounds |
| hgStar hV hSV hL hr hrhalf hLr hq0 (max 9 (K+3)) |
| obtain ⟨B,hB,hcauchy⟩ := exists_bounded_patch_cauchy_bound G d kappa q0 r Cmetric M A |
| hG hd hkappa hM hr hrhalf hCmetric hq0 hLr hrsmall K |
| refine ⟨B,hB,?_⟩ |
| intro N delta hdelt |
| have hwidth : ∀ᶠ tau : ℕ in atTop, 1 ≤ (tau : ℝ) ∧ delta/(2*(tau : ℝ)) ≤ r/4 := |
| (tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually |
| (pulse_width_eventually hr delta) |
| filter_upwards [hcutK N delta hdelt,hwidth, |
| eventually_ge_atTop (max 1 (max 9 (K+3)))] with tau hcutTau hwidthTau htau |
| intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ x hx n hn |
| have htau1 : 1 ≤ tau := (le_max_left _ _).trans htau |
| have htauk : max 9 (K+3) ≤ tau := (le_max_right _ _).trans htau |
| have htr : (0 : ℝ) < tau := zero_lt_one.trans_le (by exact_mod_cast htau1) |
| have hdw : delta/(tau : ℝ) ≤ r := by |
| have he : delta/(tau : ℝ) = 2*(delta/(2*(tau : ℝ))) := by ring |
| rw [he] |
| linarith [hwidthTau.2] |
| have hb0 : -delta/(tau : ℝ) ≤ 0 := div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hdelt.le) htr.le |
| have hab : -r ≤ -delta/(tau : ℝ) := by simpa only [neg_div] using neg_le_neg hdw |
| have hSU : modelSquare ⊆ U := hf.squareSubset.trans hf.domainSubset |
| have hfullMetric := hmetricK N delta hdelt tau htau1 htauk |
| (perturbedMetric g0 eta.val) U hg hU hSU happ |
| have hmetric : ∀ i j, CoordinateBound |
| (fun p => metricInShearCoordinates (perturbedMetric g0 eta.val) q0 p i j) |
| (shrinkingSlab (-(L*r)) (L*r) (-r) (-delta/(tau : ℝ)) (L/4)) (max 9 (K+3)) Cmetric := by |
| intro i j ds hds p hp |
| have haxes := shrinking_class_slab_axis_bounds hr.le hb0 hp |
| exact hfullMetric i j ds hds p ⟨haxes.1,hp.1⟩ |
| have hactualCut : ∀ ds : List (Fin 2), ds.length ≤ max 8 (K+2) → |
| ∀ x : ℝ, |x| ≤ L*r → |
| |iteratedCoordPartial ds (heightInShearCoordinates z q0) (coordinatePoint x (-r))| ≤ (A : ℝ) := |
| hcutTau (perturbedMetric g0 eta.val) U W z Y hg hU hgB hdet hh hf hcenter happ |
| exact hcauchy g0 eta z (-delta/(tau : ℝ)) hclass hab hb0 hgB hdet hcenter hmetric hactualCut x hx n hn |
|
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| end SmoothLocal.Perturbation |
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| end |
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| end OAI |
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