| import OAI.Geometry.IsometricImmersion.Caps.AllOrderCapEstimate |
| import OAI.Geometry.IsometricImmersion.Caps.PhysicalCapTransport |
| import OAI.Geometry.IsometricImmersion.Estimates.ActualRemainderRegularity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model SmoothLocal.HighEquation |
|
|
| theorem cap_closed_image_properties {Y : ℝ → ℝ → ℝ} |
| (hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2)) |
| (hdisp : ∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1:ℝ)/50) |
| {tl tr sb st : ℝ} (hbox : closedRectangle tl tr sb st ⊆ capChartDomain) : |
| IsCompact (capChart Y '' closedRectangle tl tr sb st) ∧ |
| MeasurableSet (capChart Y '' closedRectangle tl tr sb st) ∧ |
| volume (capChart Y '' closedRectangle tl tr sb st) < ⊤ ∧ |
| capChart Y '' closedRectangle tl tr sb st ⊆ modelOpenSquare := by |
| have hc : IsCompact (capChart Y '' closedRectangle tl tr sb st) := |
| (capRectangle_isCompact _ _ _ _).image_of_continuousOn |
| ((capChart_contDiffOn hYs).continuousOn.mono hbox) |
| refine ⟨hc,hc.measurableSet,hc.measure_lt_top,?_⟩ |
| rintro _ ⟨p,hp,rfl⟩ |
| exact capChart_mem_modelOpenSquare (hbox hp) (hdisp p (hbox hp)) |
|
|
| theorem exists_C8_one_flow_physical_cap_estimates |
| {g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord} {G Z d c e0 kappa : ℝ} |
| (hg : SmoothPositiveOn (g0+eta) U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hz : ContDiffOn ℝ ∞ z 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) |
| (hzB : ∀ k ≤ 8, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k z p‖ ≤ Z) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |((g0+eta) p).det|) |
| (hyy : ∀ p ∈ modelSquare, c ≤ |covHessian (g0+eta) z p 1 1|) |
| (hEfloor : ∀ p ∈ modelSquare, e0 ≤ heightEnergy (g0+eta) z p) |
| (hsmall : ∀ p ∈ modelSquare, |hessianQuotient (g0+eta) z p| ≤ (1:ℝ)/100) |
| (hD : ∀ p ∈ modelSquare, |
| (covHessian (g0+eta) z p).det = gaussianCurvature (g0+eta) p*heightEnergy (g0+eta) z p) |
| (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) : |
| ∃ (V : Set Coord) (Y : ℝ → ℝ → ℝ), |
| IsOpen V ∧ modelSquare ⊆ V ∧ V ⊆ U ∧ |
| ContDiffOn ℝ ∞ (hessianQuotient (g0+eta) z) V ∧ |
| CoordinateBound (hessianQuotient (g0+eta) z) modelSquare 3 (heightQuotientJetBound G Z d c) ∧ |
| ContDiffOn ℝ ∞ (capChart Y) capChartDomain ∧ |
| ContinuousOn (uncurry Y) (Icc (-2:ℝ) 2 ×ˢ Icc (-2:ℝ) 2) ∧ |
| ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2) ∧ |
| (∀ s ∈ Ioo (-2:ℝ) 2, ∀ t ∈ Ioo (-2:ℝ) 2, 0 < deriv (fun r => Y r t) s) ∧ |
| (∀ 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) ∧ |
| (∀ s ∈ Icc (-2:ℝ) 2, ∀ t ∈ Icc (-2:ℝ) 2, Y s t ∈ Icc (-3:ℝ) 3) ∧ |
| (∀ s ∈ Icc (-2:ℝ) 2, ∀ t ∈ Icc (-2:ℝ) 2, |Y s t-s| ≤ (1:ℝ)/50) ∧ |
| (∀ p ∈ capChartDomain, |capFlowHeight Y p-p 1| ≤ (1:ℝ)/50) ∧ |
| (∀ tl tr sb st : ℝ, closedRectangle tl tr sb st ⊆ capChartDomain → |
| IsCompact (capChart Y '' closedRectangle tl tr sb st) ∧ |
| MeasurableSet (capChart Y '' closedRectangle tl tr sb st) ∧ |
| volume (capChart Y '' closedRectangle tl tr sb st) < ⊤ ∧ |
| capChart Y '' closedRectangle tl tr sb st ⊆ modelOpenSquare) ∧ |
| ∀ m : ℕ, |
| (∀ p ∈ capChartDomain, |
| multiplierOperator (heightChartA (g0+eta) z Y) (heightChartB (g0+eta) z Y (m+3)) |
| (heightChartC (g0+eta) z Y (m+3)) (capPullback Y (verticalJet z (m+3))) p = |
| capPullback Y (actualHighRemainder (g0+eta) z m) p) ∧ |
| ∀ L R bottom top b : ℝ, |
| -2 < L → R < 2 → -2 < bottom → L ≤ R → bottom ≤ top → top < b → b ≤ 0 → |
| 0 < heightPhysicalCapConstant G Z d c e0 kappa (m+3) L R bottom top b ∧ |
| (∫ p in capChart Y '' closedRectangle L R bottom top, |
| (coordPartial 0 (verticalJet z (m+3)) p)^2+(verticalJet z (m+4) p)^2) ≤ |
| heightPhysicalCapConstant G Z d c e0 kappa (m+3) L R bottom top b* |
| ((∫ p in capChart Y '' closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b, |
| (actualHighRemainder (g0+eta) z m p)^2)+ |
| (∫ p in capChart Y '' closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b, |
| (verticalJet z (m+3) p)^2)) := by |
| obtain ⟨V,Y,hV,hSV,hVU,hq,hqB,hchart,hY,hYs,hvar,hstart,hode,hconf,hdisp50,hdisp,hall⟩ := |
| exists_C8_one_flow_all_high_cap_estimates hg hU hSU hz hG hZ hd hc he0 hgB hzB hdet hyy hEfloor |
| hsmall hD hk hbackground hsupport hcentral |
| obtain ⟨_,hsource,_,heq,_,_⟩ := |
| exists_global_triangularFlow_chart hq hV hSV hY hconf hstart hode |
| let M := heightQuotientJetBound G Z d c |
| have hM : 0 ≤ M := heightQuotientJetBound_nonneg hG hZ hd hc |
| have hMq (p : Coord) (hp : p ∈ modelSquare) : |
| |coordPartial 1 (hessianQuotient (g0+eta) z) p| ≤ M := hqB [1] (by norm_num) p hp |
| have hJ (p : Coord) (hp : p ∈ capChartDomain) := |
| capChart_jacobian_bounds hq hV hSV hY hconf hstart hode hM hMq hp |
| have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU |
| have hgO : SmoothPositiveOn (g0+eta) modelOpenSquare := |
| ⟨fun i j => (hg.1 i j).mono hOU,fun p hp => hg.2 p (hOU hp)⟩ |
| have hzO := hz.mono hOU |
| have hmapO : MapsTo (capChart Y) capChartDomain modelOpenSquare := |
| fun p hp => capChart_mem_modelOpenSquare hp (hdisp p hp) |
| have hyyO : ∀ p ∈ modelOpenSquare, covHessian (g0+eta) z p 1 1 ≠ 0 := |
| fun p hp => LowQuotient.denominator_ne_zero hc hyy p (modelOpenSquare_subset hp) |
| have hsmall1 (p : Coord) (hp : p ∈ capChartDomain) : |
| |hessianQuotient (g0+eta) z (capChart Y p)| ≤ 1 := |
| (hsmall _ (modelOpenSquare_subset (hmapO hp))).trans (by norm_num) |
| refine ⟨V,Y,hV,hSV,hVU,hq,hqB,hchart,hY,hYs,hvar,hstart,hode,hconf,hdisp50,hdisp, |
| (fun tl tr sb st hbox => cap_closed_image_properties hYs hdisp hbox),?_⟩ |
| intro m |
| obtain ⟨_,_,_,_,_,hPDE,hestimate⟩ := hall m |
| refine ⟨hPDE,?_⟩ |
| intro L R bottom top b hL hR hbottom hLR hbt htop hb |
| have hT : closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b ⊆ capChartDomain := |
| capOuterRectangle_subset_domain hL hR hbottom (by linarith) |
| have hS : closedRectangle L R bottom top ⊆ capChartDomain := by |
| intro p hp |
| exact ⟨⟨hL.trans_le hp.1.1,hp.1.2.trans_lt hR⟩, |
| ⟨hbottom.trans_le hp.2.1,by linarith [hp.2.2]⟩⟩ |
| obtain ⟨hlo,hli,hlu,hri,hrr,hro,hao,hai,hau,hlx,hrx⟩ := capOuterRectangle_edges hL hR hbottom |
| have hOT : capOuterLeft L ≤ capOuterRight R := by linarith |
| have hOS : capOuterBottom bottom ≤ b := by linarith |
| have hu := verticalJet_contDiffOn modelOpenSquare_isOpen hzO (m+3) |
| have hf := actualHighRemainder_contDiffOn hgO modelOpenSquare_isOpen hzO |
| (fun p hp => hD p (modelOpenSquare_subset hp)) hyyO m |
| have hfI : ContinuousOn (actualHighRemainder (g0+eta) z m) |
| (capChart Y '' closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b) := |
| hf.continuousOn.mono (by rintro _ ⟨p,hp,rfl⟩; exact hmapO (hT hp)) |
| have hcap := hestimate L R bottom top b hL hR hbottom hLR hbt htop hb |
| rw [heightUniformCapEstimateRHS_eq_factors] at hcap |
| have hp := cap_gradient_estimate_to_physical_images hYs hvar hode hsource heq hu |
| modelOpenSquare_isOpen hmapO hLR hbt hOT hOS hS hT hsmall1 hJ hfI hcap |
| refine ⟨heightPhysicalCapConstant_pos G Z d c e0 kappa (m+3) L R bottom top b,?_⟩ |
| change (∫ p in capChart Y '' closedRectangle L R bottom top, |
| coordinateGradientSquare (verticalJet z (m+3)) p) ≤ _ |
| exact hp |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
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| end OAI |
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