| import OAI.Geometry.IsometricImmersion.Caps.LowerCapInduction |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function MeasureTheory |
| open scoped ContDiff Topology BigOperators ENNReal NNReal |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model |
| open SmoothLocal.HighEquation SmoothLocal.Analytic SmoothLocal.Sobolev |
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| theorem fixed_metric_lower_cap_all_derivatives |
| {g0 eta : MetricField} {U : Set Coord} {G Z d c e0 kappa : ℝ} |
| (hg : SmoothPositiveOn (g0+eta) U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) |
| (hgB : ∀ i j : Fin 2, ∀ k ≤ 8, ∀ p ∈ modelSquare, |
| ‖iteratedFDeriv ℝ k (fun q => (g0+eta) q i j) p‖ ≤ G) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |((g0+eta) p).det|) |
| (hk : 0 < kappa) (hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p) |
| (hsupport : tsupport eta ⊆ patchBox) |
| (hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0+eta) p < -kappa/2) : |
| ∃ (B : (bStar : ℝ) → ℕ → LowerCapRectangle bStar → ℝ) |
| (H : (bStar : ℝ) → ℕ → LowerCapRectangle bStar → ℝ≥0), |
| (∀ (bStar : ℝ) (n : ℕ) (r : LowerCapRectangle bStar), 0 ≤ B bStar n r) ∧ |
| ∀ z : Coord → ℝ, ContDiffOn ℝ ∞ z U → |
| (∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z) → |
| (∀ p ∈ modelSquare, c ≤ |covHessian (g0+eta) z p 1 1|) → |
| (∀ p ∈ modelSquare, e0 ≤ heightEnergy (g0+eta) z p) → |
| (∀ p ∈ modelSquare, |hessianQuotient (g0+eta) z p| ≤ (1 : ℝ)/100) → |
| (∀ p ∈ modelSquare, |
| (covHessian (g0+eta) z p).det = gaussianCurvature (g0+eta) p*heightEnergy (g0+eta) z p) → |
| ∃ Y : ℝ → ℝ → ℝ, |
| ContDiffOn ℝ ∞ (capChart Y) capChartDomain ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s) ∧ |
| (∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| HasDerivWithinAt (Y s) (-hessianQuotient (g0+eta) z (coordinatePoint t (Y s t))) |
| (Icc (-2 : ℝ) 2) t) ∧ |
| (∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1 : ℝ)/50) ∧ |
| ∀ (bStar : ℝ) (n : ℕ) (r : LowerCapRectangle bStar), |
| CoordinateBound z (r.image Y) n (B bStar n r) ∧ |
| CoordinateL2Bound z (r.image Y) n (H bStar n r) := by |
| classical |
| have hg4 : ∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare, |
| ‖iteratedFDeriv ℝ k (fun q => (g0+eta) q i j) p‖ ≤ G := |
| fun i j k hk p hp => hgB i j k (by omega) p hp |
| have hpoint : ∀ (bStar : ℝ) (n : ℕ) (r : LowerCapRectangle bStar), |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (W : Set Coord), |
| CapInductionHeight (g0+eta) U Z c e0 z → |
| CapInductionFlow (g0+eta) U G Z d c e0 kappa z Y W → CoordinateBound z (r.image Y) n B := |
| fun _ n r => uniform_lowerCap_pointwise hg hU hSU hG hZ hd hc hg4 hdet n r |
| choose B hB hBest using hpoint |
| have hnorm : ∀ (bStar : ℝ) (n : ℕ) (r : LowerCapRectangle bStar), |
| ∃ H : ℝ≥0, ∀ (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (W : Set Coord), |
| CapInductionHeight (g0+eta) U Z c e0 z → |
| CapInductionFlow (g0+eta) U G Z d c e0 kappa z Y W → CoordinateL2Bound z (r.image Y) (n+8) H := |
| fun _ n r => uniform_lowerCap_L2 hg hU hSU hG hZ hd hc hg4 hdet n r |
| choose H hHest using hnorm |
| refine ⟨B,H,hB,?_⟩ |
| intro z hz hzB hyy hE hsmall hD |
| let hh : CapInductionHeight (g0+eta) U Z c e0 z := ⟨hz,hzB,hyy,hE,hsmall,hD⟩ |
| obtain ⟨W,Y,hf⟩ := exists_capInductionFlow hg hU hSU hG hZ hd hc he0 hgB hdet hh |
| hk hbackground hsupport hcentral |
| refine ⟨Y,capChart_contDiffOn hf.jointSmooth,hf.start,hf.ode,hf.displacement,?_⟩ |
| intro bStar n r |
| exact ⟨hBest bStar n r z Y W hh hf, |
| (hHest bStar n r z Y W hh hf).mono (by omega) le_rfl⟩ |
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| end SmoothLocal.Flow |
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| end |
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| end OAI |
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