| import OAI.Geometry.IsometricImmersion.Caps.C8ActualCapEstimate |
| import OAI.Geometry.IsometricImmersion.Caps.CapFlowUniqueness |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model SmoothLocal.HighEquation |
|
|
| theorem exists_C8_one_flow_all_high_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) ∧ |
| ∀ m : ℕ, |
| let epsilon := heightDirectedEpsilon G Z d c e0 kappa (m+3) |
| let lambda := heightDirectedLambda G Z d c e0 kappa (m+3) |
| let c1 := heightDirectedSlope G Z d c e0 kappa (m+3) |
| let c2 := heightDirectedCoercivity G Z d c e0 kappa (m+3) |
| 0 < epsilon ∧ epsilon ≤ 1 ∧ 0 < lambda ∧ 0 < c1 ∧ 0 < c2 ∧ |
| (∀ 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 → |
| rectangleIntegral L R bottom top |
| (fun p => (coordPartial 0 (capPullback Y (verticalJet z (m+3))) p)^2+ |
| (coordPartial 1 (capPullback Y (verticalJet z (m+3))) p)^2) ≤ |
| heightUniformCapEstimateRHS G Z d c e0 kappa (m+3) L R bottom top b |
| (capPullback Y (verticalJet z (m+3))) |
| (capPullback Y (actualHighRemainder (g0+eta) z m)) := by |
| have hg4 (i j : Fin 2) : CoordinateBound (fun p => (g0+eta) p i j) modelSquare 4 G := |
| (coordinateBound_of_frechet_bounds (hg.1 i j) hU hSU (hgB i j)).mono (by norm_num) le_rfl |
| have hz5 : CoordinateBound z modelSquare 5 Z := coordinateBound_five_of_frechet_eight hz hU hSU hzB |
| obtain ⟨V,hV,hSV,hVU,_,hq,hqB⟩ := |
| exists_open_quotient_domain_with_closed_low_bounds hg hU hSU hz hG hZ hd hc hg4 hz5 hdet hyy |
| obtain ⟨Y,hY,_,hstart,hode,_,hdisp50,hrange,_⟩ := exists_cap_flow hq hV hSV hsmall |
| have hconf (s : ℝ) (hs : s ∈ Icc (-2:ℝ) 2) (t : ℝ) (ht : t ∈ Icc (-2:ℝ) 2) : |
| Y s t ∈ Icc (-3:ℝ) 3 := abs_le.mp ((hrange s hs t ht).1.trans (by norm_num)) |
| have hYs := cap_flow_joint_contDiffOn hq hV hSV hY hconf hstart hode |
| have hchart := capChart_contDiffOn hYs |
| have hvar : ∀ s ∈ Ioo (-2:ℝ) 2, ∀ t ∈ Ioo (-2:ℝ) 2, 0 < deriv (fun r => Y r t) s := |
| fun s hs t ht => cap_flow_initial_deriv_pos hq hV hSV hY hconf hstart hode hs ht |
| have hdisp (p : Coord) (hp : p ∈ capChartDomain) : |capFlowHeight Y p-p 1| ≤ (1:ℝ)/50 := |
| hdisp50 (p 1) ⟨hp.2.1.le,hp.2.2.le⟩ (p 0) ⟨hp.1.1.le,hp.1.2.le⟩ |
| 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 hPDE (m : ℕ) (p : Coord) (hp : 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 := |
| actual_flow_high_equation hgO modelOpenSquare_isOpen hzO |
| (fun p hp => hD p (modelOpenSquare_subset hp)) hyyO hYs hode hvar hmapO m hp |
| refine ⟨V,Y,hV,hSV,hVU,hq,hqB,hchart,hY,hYs,hvar,hstart,hode,hconf,hdisp50,hdisp,?_⟩ |
| intro m |
| obtain ⟨Ym,heps,heps1,hlambda,hc1,hc2,_,hstartm,hodem,hdispm,_,hestimatem⟩ := |
| exists_uniform_original_data_actual_high_cap_estimate hg hU hSU hz hG hZ hd hc he0 hg4 hz5 |
| hdet hyy hEfloor hsmall hD hk hbackground hsupport hcentral m |
| have heq := capChart_eqOn_of_same_ode hq hV hSV hstart hode hdisp hstartm hodem hdispm |
| refine ⟨heps,heps1,hlambda,hc1,hc2,hPDE m,?_⟩ |
| 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 huint := capPullback_square_rectangle_eq_of_chart_eqOn heq (verticalJet z (m+3)) hOT hOS hT |
| have hfint := capPullback_square_rectangle_eq_of_chart_eqOn heq (actualHighRemainder (g0+eta) z m) hOT hOS hT |
| have hRHS : heightUniformCapEstimateRHS G Z d c e0 kappa (m+3) L R bottom top b |
| (capPullback Y (verticalJet z (m+3))) (capPullback Y (actualHighRemainder (g0+eta) z m)) = |
| heightUniformCapEstimateRHS G Z d c e0 kappa (m+3) L R bottom top b |
| (capPullback Ym (verticalJet z (m+3))) (capPullback Ym (actualHighRemainder (g0+eta) z m)) := by |
| unfold heightUniformCapEstimateRHS |
| rw [huint,hfint] |
| rw [capPullback_gradient_rectangle_eq_of_chart_eqOn heq (verticalJet z (m+3)) hLR hbt hS,hRHS] |
| exact hestimatem L R bottom top b hL hR hbottom hLR hbt htop hb |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
