| import OAI.Geometry.IsometricImmersion.Flows.FlowHeightCoefficients |
| import OAI.Geometry.IsometricImmersion.Obstructions.PatchAdmissibility |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Function |
| open scoped ContDiff |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
|
|
| def modelOpenSquare : Set Coord := coordinateRectangle 3 (-3) 3 |
|
|
| theorem modelOpenSquare_isOpen : IsOpen modelOpenSquare := coordinateRectangle_isOpen _ _ _ |
|
|
| theorem modelOpenSquare_subset : modelOpenSquare ⊆ modelSquare := by |
| intro p hp |
| constructor <;> intro i <;> fin_cases i |
| · exact hp.1.1.le |
| · exact hp.2.1.le |
| · exact hp.1.2.le |
| · exact hp.2.2.le |
|
|
| theorem capChart_mem_modelOpenSquare |
| (hp : p ∈ capChartDomain) (hdisp : |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) : |
| capChart Y p ∈ modelOpenSquare := by |
| change (capChart Y p) 0 ∈ Ioo (-3 : ℝ) 3 ∧ (capChart Y p) 1 ∈ Ioo (-3 : ℝ) 3 |
| rw [capChart_zero, capChart_one] |
| constructor |
| · constructor <;> linarith [hp.1.1, hp.1.2] |
| · constructor <;> linarith [hp.2.1, hp.2.2, (abs_le.mp hdisp).1, (abs_le.mp hdisp).2] |
|
|
| theorem exists_admissible_height_operator_chart |
| (h : PatchAdmissibleHeight g z) : |
| ∃ Y : ℝ → ℝ → ℝ, |
| ContDiffOn ℝ ∞ (capChart Y) capChartDomain ∧ |
| MapsTo (capChart Y) capChartDomain modelOpenSquare ∧ |
| (∀ p ∈ capChartDomain, |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) ∧ |
| ContDiffOn ℝ ∞ (capChartRho Y) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (capChartSigma Y) capChartDomain ∧ |
| (∀ p ∈ capChartDomain, 0 < capChartRho Y p ∧ 0 < heightChartG1 g z Y p) ∧ |
| (∀ ell : ℕ, ContDiffOn ℝ ∞ (heightChartC g z Y ell) capChartDomain ∧ |
| ContDiffOn ℝ ∞ (heightChartRemainder g z Y ell) capChartDomain ∧ |
| ∀ p ∈ capChartDomain, |
| heightChartC g z Y ell p = ((ell : ℝ) + 1) * coordPartial 1 (heightChartA g z Y) p + |
| heightChartA g z Y p * heightChartRemainder g z Y ell p) ∧ |
| (∀ u : Coord → ℝ, ContDiffOn ℝ ∞ u modelOpenSquare → ∀ p ∈ capChartDomain, |
| coordPartial 0 (capPullback Y u) p = |
| coordinateDrift (hessianQuotient g z) u (capChart Y p) ∧ |
| capPullback Y (coordPartial 1 u) p = capChartRho Y p * coordPartial 1 (capPullback Y u) p ∧ |
| capPullback Y (coordPartial 1 (coordPartial 1 u)) p = |
| (capChartRho Y p)^2 * coordPartial 1 (coordPartial 1 (capPullback Y u)) p + |
| capChartSigma Y p * coordPartial 1 (capPullback Y u) p ∧ |
| ∀ ell : ℕ, |
| capPullback Y (driftOperator (hessianQuotient g z) (heightOriginalA g z) |
| (heightOriginalB g z ell) (heightOriginalC0 g z ell) u) p = |
| multiplierOperator (heightChartA g z Y) (heightChartB g z Y ell) |
| (heightChartC g z Y ell) (capPullback Y u) p) := by |
| obtain ⟨Y, hY, hstart, hode, hdisp, hvar⟩ := patchAdmissibleHeight_exists_smooth_flow h |
| obtain ⟨U, hU, hSU, hg, hz, hD, hE, hyy, hsmall⟩ := h |
| have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU |
| have hgo : SmoothPositiveOn g modelOpenSquare := |
| ⟨fun i j => (hg.1 i j).mono hOU, fun p hp => hg.2 p (hOU hp)⟩ |
| have hzo : ContDiffOn ℝ ∞ z modelOpenSquare := hz.mono hOU |
| have hDo : ∀ p ∈ modelOpenSquare, |
| (covHessian g z p).det = gaussianCurvature g p * heightEnergy g z p := |
| fun p hp => hD p (modelOpenSquare_subset hp) |
| have hEo : ∀ p ∈ modelOpenSquare, 0 < heightEnergy g z p := |
| fun p hp => hE p (modelOpenSquare_subset hp) |
| have hyyo : ∀ p ∈ modelOpenSquare, covHessian g z p 1 1 ≠ 0 := |
| fun p hp => hyy p (modelOpenSquare_subset hp) |
| have hvarpos : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, |
| 0 < deriv (fun r => Y r t) s := fun s hs t ht => (hvar s hs t ht).2 |
| have hdisp' : ∀ p ∈ capChartDomain, |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50 := by |
| intro p hp |
| exact hdisp (p 1) ⟨hp.2.1.le, hp.2.2.le⟩ (p 0) ⟨hp.1.1.le, hp.1.2.le⟩ |
| have hmap : MapsTo (capChart Y) capChartDomain modelOpenSquare := |
| fun p hp => capChart_mem_modelOpenSquare hp (hdisp' p hp) |
| refine ⟨Y, capChart_contDiffOn hY, hmap, hdisp', capChartRho_contDiffOn hY hvarpos, |
| capChartSigma_contDiffOn hY hvarpos, ?_, ?_, ?_⟩ |
| · intro p hp |
| exact ⟨capChartRho_pos hY hvarpos hp, |
| heightChartG1_pos g z hY hvarpos hp (hEo _ (hmap hp)) (hyyo _ (hmap hp))⟩ |
| · intro ell |
| exact ⟨heightChartC_contDiffOn hgo modelOpenSquare_isOpen hzo hDo hyyo hEo hY hvarpos hmap ell, |
| heightChartRemainder_contDiffOn hgo modelOpenSquare_isOpen hzo hyyo hEo hY hvarpos hmap ell, |
| fun p hp => heightChartC_factor hgo modelOpenSquare_isOpen hzo hDo hyyo hEo hY hvarpos hmap hp ell⟩ |
| · intro u hu p hp |
| have hud : DifferentiableAt ℝ u (capChart Y p) := |
| ((hu.contDiffAt (modelOpenSquare_isOpen.mem_nhds (hmap hp))).differentiableAt (by simp)) |
| exact ⟨capPullback_partial_time hY hode hp hud, |
| capPullback_original_y hY hvarpos hp hud, |
| capPullback_original_yy hY hvarpos hu modelOpenSquare_isOpen hmap hp, |
| fun ell => height_coefficient_operator_transport hgo modelOpenSquare_isOpen hzo hyyo hY hode hvarpos hmap hu hp ell⟩ |
| |
| end SmoothLocal.Flow |
| |
| end |
| |
| end OAI |
| |
