| import OAI.Geometry.IsometricImmersion.Caps.PulseCapSeparation |
| import OAI.Geometry.IsometricImmersion.Darboux.QSegmentDomain |
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|
| namespace OAI |
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|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE SmoothLocal.Weighted |
| open SmoothLocal.HighEquation |
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|
| theorem metric_jets_eq_on_actual_qJetSegment |
| {g h : MetricField} {p : Coord} (heq : g =ᶠ[𝓝 p] h) |
| (z0 z : Coord → ℝ) (sigma : ℝ) (n : ℕ) (i j : Fin 2) : |
| iteratedFDeriv ℝ n (fun v => g v i j) (statePoint (qHeightJetSegment z0 z sigma p)) = |
| iteratedFDeriv ℝ n (fun v => h v i j) (statePoint (qHeightJetSegment z0 z sigma p)) := by |
| rw [statePoint_qHeightJetSegment] |
| exact ((metric_coeff_eventuallyEq heq i j).iteratedFDeriv ℝ n).self_of_nhds |
|
|
| theorem exists_uniform_actual_pulse_cap_exclusion |
| {G Z d c e0 kappa q0 L : ℝ} |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) |
| (hL : 0 < L) (hq0 : |q0| ≤ 1/20) : |
| ∃ r : ℝ, 0 < r ∧ r < 1/2 ∧ L*r ≤ 1/20 ∧ |
| ∀ N : ℕ, ∀ delta : ℝ, 0 < delta → |
| ∀ᶠ tau : ℝ in atTop, 1 ≤ tau ∧ |
| ∀ (g gStar : MetricField) (U W : Set Coord) (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ), |
| CapInductionHeight g U Z c e0 z → CapInductionFlow g U G Z d c e0 kappa z Y W → |
| |hessianQuotient g z 0-q0| ≤ 1/(100*L) → |
| ∀ R : LowerCapRectangle (-r/2), ∀ p ∈ R.image Y, |
| testMetric gStar q0 (L*r/16) N delta tau =ᶠ[𝓝 p] gStar ∧ |
| ∀ (z0 : Coord → ℝ) (sigma : ℝ) (n : ℕ) (i j : Fin 2), |
| iteratedFDeriv ℝ n (fun v => testMetric gStar q0 (L*r/16) N delta tau v i j) |
| (statePoint (qHeightJetSegment z0 z sigma p)) = |
| iteratedFDeriv ℝ n (fun v => gStar v i j) |
| (statePoint (qHeightJetSegment z0 z sigma p)) := by |
| let M := heightQuotientJetBound G Z d c |
| have hM : 0 ≤ M := heightQuotientJetBound_nonneg hG hZ hd hc |
| obtain ⟨r,hr,hrhalf,hLr,hrsmall⟩ := exists_sectionFour_radius hM hL |
| refine ⟨r,hr,hrhalf,hLr,?_⟩ |
| intro N delta hdelta |
| filter_upwards [pulse_width_eventually hr delta] with tau htau |
| refine ⟨htau.1,?_⟩ |
| intro g gStar U W z Y hh hf hcenter R p hp |
| have hpartial (v : Coord) (hv : v ∈ modelSquare) (i : Fin 2) : |
| |coordPartial i (hessianQuotient g z) v| ≤ M := hf.quotientBound [i] (by norm_num) v hv |
| have hLrpos : 0 < L*r := mul_pos hL hr |
| have heq := testMetric_eventuallyEq_on_lower_cap hf.quotientSmooth hf.domainOpen hf.squareSubset |
| hf.continuous hf.range hf.start hf.ode hh.quotient hM hL hr hpartial hq0 hcenter |
| (by linarith : L*r < 2) (by positivity : 0 < L*r/16) (by linarith : L*r/16 ≤ L*r) |
| hrsmall hdelta (by linarith [htau.1] : 0 < tau) htau.2 gStar N R hp |
| refine ⟨heq,?_⟩ |
| intro z0 sigma n i j |
| exact metric_jets_eq_on_actual_qJetSegment heq z0 z sigma n i j |
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| end SmoothLocal.Pulse |
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| end |
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| end OAI |
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