| import OAI.Geometry.IsometricImmersion.Metrics.UniversalMetricQ |
| import OAI.Geometry.IsometricImmersion.Metrics.LocalMetricPCoefficients |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry |
|
|
| theorem metricPBundle_mapsTo_Q_domain {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) : |
| MapsTo (metricPBundle g) (darbouxQStateDomain g U) universalMetricQDomain := by |
| intro w hw |
| exact ⟨metricDet_ne_zero hg hw.1, |
| by simpa only [universalQDenominator_metricPBundle] using hw.2⟩ |
|
|
| theorem metricPBundle_mem_universalQTube {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {w : DarbouxState} |
| (hw : statePoint w ∈ U) {G W d c : ℝ} (hG : 0 ≤ G) |
| (hgB : ∀ i j k, k ≤ 2 → |
| ‖iteratedFDeriv ℝ k (fun p => g p i j) (statePoint w)‖ ≤ G) |
| (hwB : ‖w‖ ≤ W) (hdet : d ≤ |(g (statePoint w)).det|) |
| (hden : c ≤ |stateQDenominator g w|) : |
| metricPBundle g w ∈ universalMetricQTube (max G W) d c := by |
| refine ⟨⟨?_, hdet⟩, ?_⟩ |
| · simpa only [Metric.mem_closedBall, dist_zero_right] using |
| metricPBundle_norm_bound hg hU hw hG hgB hwB |
| · simpa only [Set.mem_preimage, Set.mem_ofPred_eq, universalQDenominator_metricPBundle] using hden |
|
|
| theorem actualQ_jet_bound_from_universalTube {g : MetricField} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) {w : DarbouxState} |
| (hw : w ∈ darbouxQStateDomain g U) {N : ℕ} {G W d c C : ℝ} |
| (hG : 0 ≤ G) (hd : 0 < d) (hc : 0 < c) |
| (hgB : ∀ i j k, k ≤ N + 2 → |
| ‖iteratedFDeriv ℝ k (fun p => g p i j) (statePoint w)‖ ≤ G) |
| (hwB : ‖w‖ ≤ W) (hdet : d ≤ |(g (statePoint w)).det|) |
| (hden : c ≤ |stateQDenominator g w|) |
| (hC : ∀ k ≤ N, ∀ a ∈ universalMetricQTube (max G W) d c, |
| ‖iteratedFDeriv ℝ k universalMetricQ a‖ ≤ C) |
| {n : ℕ} (hn : n ≤ N) : |
| ‖iteratedFDeriv ℝ n (sixVariableQ g) w‖ ≤ n.factorial * C * (max 1 G)^n := by |
| have hD := darbouxQStateDomain_isOpen hg hU |
| have hsub : darbouxQStateDomain g U ⊆ stateBaseDomain U := fun _ hv => hv.1 |
| have hf := (metricPBundle_contDiffOn hg hU).mono hsub |
| have hmap := metricPBundle_mapsTo_Q_domain hg |
| have hmem := metricPBundle_mem_universalQTube hg hU hw.1 hG |
| (fun i j k hk => hgB i j k (by omega)) hwB hdet hden |
| have ho : ∀ i, i ≤ n → |
| ‖iteratedFDerivWithin ℝ i universalMetricQ universalMetricQDomain (metricPBundle g w)‖ ≤ C := by |
| intro i hi |
| rw [iteratedFDerivWithin_of_isOpen i universalMetricQDomain_isOpen |
| (universalMetricQTube_subset_domain (max G W) hd hc hmem)] |
| exact hC i (hi.trans hn) (metricPBundle g w) hmem |
| have hi : ∀ i, 1 ≤ i → i ≤ n → |
| ‖iteratedFDerivWithin ℝ i (metricPBundle g) (darbouxQStateDomain g U) w‖ ≤ (max 1 G)^i := by |
| intro i hi hin |
| rw [iteratedFDerivWithin_of_isOpen i hD hw] |
| exact (metricPBundle_positive_jet_bound hg hU hw.1 hG hgB hi (hin.trans hn)).trans |
| (le_self_pow₀ (le_max_left _ _) (by omega)) |
| have hcomp := norm_iteratedFDerivWithin_comp_le universalMetricQ_contDiffOn hf |
| (WithTop.coe_le_coe.mpr le_top) universalMetricQDomain_isOpen.uniqueDiffOn hD.uniqueDiffOn |
| hmap hw ho hi |
| have heq : iteratedFDerivWithin ℝ n (sixVariableQ g) (darbouxQStateDomain g U) w = |
| iteratedFDerivWithin ℝ n (universalMetricQ ∘ metricPBundle g) (darbouxQStateDomain g U) w := |
| iteratedFDerivWithin_congr (fun _ hv => sixVariableQ_eq_universalMetricQ hg hU hv.1) hw n |
| rw [iteratedFDerivWithin_of_isOpen n hD hw] at heq |
| rw [← heq] at hcomp |
| exact hcomp |
|
|
| theorem exists_local_metric_Q_jet_bound (G W : ℝ) (hG : 0 ≤ G) |
| {d c : ℝ} (hd : 0 < d) (hc : 0 < c) (N : ℕ) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (g : MetricField) (U : Set Coord), |
| SmoothPositiveOn g U → IsOpen U → ∀ w : DarbouxState, |
| statePoint w ∈ U → |
| (∀ i j k, k ≤ N + 2 → |
| ‖iteratedFDeriv ℝ k (fun p => g p i j) (statePoint w)‖ ≤ G) → |
| ‖w‖ ≤ W → d ≤ |(g (statePoint w)).det| → c ≤ |stateQDenominator g w| → |
| ∀ n ≤ N, ‖iteratedFDeriv ℝ n (sixVariableQ g) w‖ ≤ C := by |
| classical |
| obtain ⟨C0, hC0, hCP⟩ := universalMetricQTube_finite_bounds (max G W) hd hc N |
| let C : ℝ := ∑ k : Fin (N + 1), (k.val.factorial : ℝ) * C0 * (max 1 G)^k.val |
| have hC : 0 ≤ C := Finset.sum_nonneg (fun _ _ => by positivity) |
| refine ⟨C, hC, ?_⟩ |
| intro g U hg hU w hw hgB hwB hdet hden n hn |
| have hne : stateQDenominator g w ≠ 0 := by |
| intro hz |
| rw [hz, abs_zero] at hden |
| linarith |
| have hb := actualQ_jet_bound_from_universalTube hg hU ⟨hw, hne⟩ hG hd hc hgB hwB hdet hden hCP hn |
| let j : Fin (N + 1) := ⟨n, by omega⟩ |
| have hsum : (j.val.factorial : ℝ) * C0 * (max 1 G)^j.val ≤ |
| ∑ i : Fin (N+1), (i.val.factorial : ℝ) * C0 * (max 1 G)^i.val := |
| Finset.single_le_sum |
| (f := fun i : Fin (N+1) => (i.val.factorial : ℝ) * C0 * (max 1 G)^i.val) |
| (fun _ _ => by positivity) (Finset.mem_univ j) |
| exact hb.trans hsum |
|
|
| end SmoothLocal.HighEquation |
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|
| end |
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| end OAI |
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