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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 | import OAI.Geometry.IsometricImmersion.Caps.LowerCapInduction
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology BigOperators ENNReal NNReal
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model
open SmoothLocal.HighEquation SmoothLocal.Analytic SmoothLocal.Sobolev
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⟩
end SmoothLocal.Flow
end
end OAI
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