File size: 5,348 Bytes
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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | 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 {Y : ℝ → ℝ → ℝ} {p : Coord}
(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
{g : MetricField} {z : Coord → ℝ} (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
|