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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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Geometry.IsometricImmersion.Metrics.LocalMetricQActive
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic
theorem qSpatialProjection_norm_le (w : DarbouxState) : ‖qSpatialProjection w‖ ≤ ‖w‖ := by
apply (pi_norm_le_iff_of_nonneg (norm_nonneg w)).mpr
intro i
fin_cases i
· exact norm_le_pi_norm w 0
· exact norm_le_pi_norm w 2
· exact norm_le_pi_norm w 3
· exact norm_le_pi_norm w 4
· exact norm_le_pi_norm w 5
theorem qSpatialCLM_projection_of_time_zero {w : DarbouxState} (hw : w 1 = 0) :
qSpatialCLM (qSpatialProjection w) = w := by
have he := qFixedTimeState_projection w
simpa only [qFixedTimeState, hw, Pi.single_zero, add_zero] using he
theorem coordPartial_qSolutionJet_xi_norm_bound
{z : Coord → ℝ} {U S : Set Coord} {Z : ℝ}
(hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U) (hSU : S ⊆ U)
(hzB : CoordinateBound z S 3 Z) {p : Coord} (hp : p ∈ S) :
‖coordPartial 0 (qSolutionJet z) p‖ ≤ max 1 Z := by
have he := coordinate_qSolutionJet_horizontal_positive hU hz (hSU hp) 1 (by norm_num)
change coordPartial 0 (qSolutionJet z) p = _ at he
rw [he]
apply (pi_norm_le_iff_of_nonneg (zero_le_one.trans (le_max_left _ _))).mpr
intro i
rw [Real.norm_eq_abs]
fin_cases i
· simp
· simp
· exact (hzB [0, 0] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hzB [0, 1] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hzB [0, 0, 1] (by norm_num) p hp).trans (le_max_right _ _)
· exact (hzB [0, 0, 0] (by norm_num) p hp).trans (le_max_right _ _)
theorem heightQCoefficient_xi_eq_fixedTime_second
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hp : p ∈ U) (hxx : covHessian g z p 0 0 ≠ 0) (i : Fin 6) (hi : i ≠ 1) :
coordPartial 0 (heightQCoefficient g z i) p =
iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p))
![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) 1] := by
have he : qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) = qSolutionJet z p :=
qFixedTimeState_projection (qSolutionJet z p)
have hv : qSpatialProjection (qSolutionJet z p) ∈ qFixedTimeDomain g U (p 1) := by
change qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) ∈ darbouxQStateDomain g U
rw [he]
exact qSolutionJet_mem_domain hp hxx
have htime : coordPartial 0 (qSolutionJet z) p 1 = 0 := by
have hh := congrArg (fun v : DarbouxState => v 1)
(coordinate_qSolutionJet_horizontal_positive hU hz hp 1 (by norm_num))
exact hh
rw [qFixedTimeQ_iteratedFDeriv hg hU hv 2, ContinuousMultilinearMap.compContinuousLinearMap_apply, he]
have hvec : (fun j : Fin 2 => qSpatialCLM
(![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) 1] j)) =
![coordPartial 0 (qSolutionJet z) p, Pi.single i 1] := by
funext j
fin_cases j
· exact qSpatialCLM_projection_of_time_zero htime
· change qSpatialCLM (Pi.single (qSpatialIndex i) 1) = Pi.single i 1
rw [qSpatialCLM_single, qSpatialAxis_index hi]
rw [hvec]
exact (second_fderiv_comp_constant_direction_xi
(sixVariableQ_contDiffAt_solutionJet hg hU hp hxx)
(((qSolutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp))
(Pi.single i 1)).symm
theorem heightQCoefficient_xi_bound_of_fixedTime_second
{g : MetricField} {z : Coord → ℝ} {U S : Set Coord} {Z C : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hSU : S ⊆ U) (hzB : CoordinateBound z S 3 Z) (hC : 0 ≤ C)
{p : Coord} (hp : p ∈ S) (hxx : covHessian g z p 0 0 ≠ 0)
(hQ : ‖iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1)) (qSpatialProjection (qSolutionJet z p))‖ ≤ C)
(i : Fin 6) (hi : i ≠ 1) :
|coordPartial 0 (heightQCoefficient g z i) p| ≤ C * max 1 Z := by
rw [heightQCoefficient_xi_eq_fixedTime_second hg hU hz (hSU hp) hxx i hi, ← Real.norm_eq_abs]
have hd := (qSpatialProjection_norm_le (coordPartial 0 (qSolutionJet z) p)).trans
(coordPartial_qSolutionJet_xi_norm_bound hU hz hSU hzB hp)
have hb := (iteratedFDeriv ℝ 2 (qFixedTimeQ g (p 1))
(qSpatialProjection (qSolutionJet z p))).le_opNorm
![qSpatialProjection (coordPartial 0 (qSolutionJet z) p), Pi.single (qSpatialIndex i) (1 : ℝ)]
simp only [Fin.prod_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Pi.norm_single, norm_one, mul_one] at hb
exact hb.trans (mul_le_mul hQ hd (norm_nonneg _) hC)
theorem exists_same_region_Q_first_xi_bound
(G R Z : ℝ) (hG : 0 ≤ G) (hR : 0 ≤ R)
{d c : ℝ} (hd : 0 < d) (hc : 0 < c) :
∃ Cfirst : ℝ, 0 ≤ Cfirst ∧ ∀ (g : MetricField) (z : Coord → ℝ) (U S : Set Coord),
SmoothPositiveOn g U → IsOpen U → ContDiffOn ℝ ∞ z U → S ⊆ U →
(∀ p ∈ S, |p 0| ≤ R ∧ |p 1| ≤ R) →
(∀ p ∈ S, HorizontalMetricTwoJetBound g p 2 G) →
CoordinateBound z S 3 Z →
(∀ p ∈ S, d ≤ |(g p).det|) →
(∀ p ∈ S, c ≤ |covHessian g z p 0 0|) →
∀ i : Fin 6, i ≠ 1 → ∀ p ∈ S, |coordPartial 0 (heightQCoefficient g z i) p| ≤ Cfirst := by
obtain ⟨C, hC, hQ⟩ := exists_local_fixedTime_Q_jet_bound G (max R Z) hG hd hc 2
refine ⟨C * max 1 Z, mul_nonneg hC (zero_le_one.trans (le_max_left _ _)), ?_⟩
intro g z U S hg hU hz hSU hcoords hgB hzB hdet hden i hi p hp
have hxx : covHessian g z p 0 0 ≠ 0 := by
intro he
have hh := hden p hp
rw [he, abs_zero] at hh
linarith
have he : qFixedTimeState (p 1) (qSpatialProjection (qSolutionJet z p)) = qSolutionJet z p :=
qFixedTimeState_projection (qSolutionJet z p)
have hpoint : coordinatePoint (qSpatialProjection (qSolutionJet z p) 0) (p 1) = p := point_eta p
have hz2 : CoordinateBound z S 2 Z := fun ds hds p hp => hzB ds (by omega) p hp
have hbound := hQ g U hg hU (p 1) (qSpatialProjection (qSolutionJet z p))
(by simpa only [hpoint] using hSU hp)
(by simpa only [hpoint] using hgB p hp)
(by rw [he]; exact qSolutionJet_norm_bound_of_coordinateBound hR hz2 hcoords hp)
(by simpa only [hpoint] using hdet p hp)
(by rw [he, stateQDenominator_qSolutionJet]; exact hden p hp) 2 le_rfl
exact heightQCoefficient_xi_bound_of_fixedTime_second hg hU hz hSU hzB hC hp hxx hbound i hi
end SmoothLocal.HighEquation
end
end OAI
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