| import OAI.Geometry.IsometricImmersion.Calculus.HessianJetBounds |
| import OAI.Geometry.IsometricImmersion.Obstructions.PatchAdmissibility |
| import Mathlib.Topology.Order.OrderClosed |
| import Mathlib.Topology.Order.DenselyOrdered |
| import Mathlib.Order.Interval.Set.Pi |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Geometry |
| open SmoothLocal.Flow SmoothLocal.ODE |
|
|
| theorem modelSquare_closure_interior : closure (interior modelSquare) = modelSquare := by |
| unfold modelSquare |
| rw [← Set.pi_univ_Icc, interior_pi_set (Set.finite_univ : (Set.univ : Set (Fin 2)).Finite)] |
| simp only [interior_Icc, closure_pi_set, |
| closure_Ioo (by norm_num : (-3 : ℝ) ≠ 3), Set.pi_univ_Icc] |
|
|
| theorem CoordinateBound.restrict_domain {f : Coord → ℝ} {S T : Set Coord} {n : ℕ} {M : ℝ} |
| (hf : CoordinateBound f S n M) (hTS : T ⊆ S) : CoordinateBound f T n M := |
| fun ds hds p hp => hf ds hds p (hTS hp) |
|
|
| theorem CoordinateBound.of_modelSquare_interior {f : Coord → ℝ} {U : Set Coord} {n : ℕ} {M : ℝ} |
| (hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hB : CoordinateBound f (interior modelSquare) n M) : CoordinateBound f modelSquare n M := by |
| intro ds hds p hp |
| have hcont : ContinuousOn (fun x => |iteratedCoordPartial ds f x|) modelSquare := |
| ((LowQuotient.iterated_contDiffOn hf hU ds).continuousOn.mono hSU).abs |
| have hcl : p ∈ closure (interior modelSquare) := by rw [modelSquare_closure_interior]; exact hp |
| apply le_on_closure (fun x hx => hB ds hds x hx) _ continuousOn_const hcl |
| rw [modelSquare_closure_interior] |
| exact hcont |
|
|
| variable {g : MetricField} {z : Coord → ℝ} {U : Set Coord} {G Z d c : ℝ} |
|
|
| theorem inverseMetric_coordinate_bound_closed_square |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hG : 0 ≤ G) (hd : 0 < d) |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) modelSquare 3 G) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) (i j : Fin 2) : |
| CoordinateBound (fun p => inverseMetric g p i j) modelSquare 3 (metricInverseJetBound G d) := by |
| have hIU : interior modelSquare ⊆ U := interior_subset.trans hSU |
| have hgI : SmoothPositiveOn g (interior modelSquare) := |
| ⟨fun a b => (hg.1 a b).mono hIU, fun p hp => hg.2 p (hIU hp)⟩ |
| have hBI := inverseMetric_coordinate_bound hgI isOpen_interior hG hd |
| (fun a b => (hgB a b).restrict_domain interior_subset) |
| (fun p hp => hdet p (interior_subset hp)) i j |
| exact CoordinateBound.of_modelSquare_interior (inverseMetric_contDiffOn hg i j) hU hSU hBI |
|
|
| theorem covHessian_coordinate_bound_closed_square |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hz : ContDiffOn ℝ ∞ z U) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) modelSquare 4 G) |
| (hzB : CoordinateBound z modelSquare 5 Z) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) (i j : Fin 2) : |
| CoordinateBound (fun p => covHessian g z p i j) modelSquare 3 (hessianJetBound G Z d) := by |
| have hIU : interior modelSquare ⊆ U := interior_subset.trans hSU |
| have hgI : SmoothPositiveOn g (interior modelSquare) := |
| ⟨fun a b => (hg.1 a b).mono hIU, fun p hp => hg.2 p (hIU hp)⟩ |
| have hBI := covHessian_coordinate_bound hgI isOpen_interior (hz.mono hIU) hG hZ hd |
| (fun a b => (hgB a b).restrict_domain interior_subset) (hzB.restrict_domain interior_subset) |
| (fun p hp => hdet p (interior_subset hp)) i j |
| exact CoordinateBound.of_modelSquare_interior (covHessian_contDiffOn hg hU hz i j) hU hSU hBI |
|
|
| theorem exists_open_quotient_domain_with_closed_low_bounds |
| (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| (hz : ContDiffOn ℝ ∞ z U) (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) |
| (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) modelSquare 4 G) |
| (hzB : CoordinateBound z modelSquare 5 Z) |
| (hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|) |
| (hyy : ∀ p ∈ modelSquare, c ≤ |covHessian g z p 1 1|) : |
| ∃ V : Set Coord, IsOpen V ∧ modelSquare ⊆ V ∧ V ⊆ U ∧ |
| (∀ p ∈ V, covHessian g z p 1 1 ≠ 0) ∧ |
| ContDiffOn ℝ ∞ (hessianQuotient g z) V ∧ |
| CoordinateBound (hessianQuotient g z) modelSquare 3 |
| (LowQuotient.boundThroughThree (hessianJetBound G Z d) c) := by |
| obtain ⟨V, hV, hSV, hVU, hVne, hqV⟩ := exists_open_hessianQuotient_domain hg hU hz hSU |
| (LowQuotient.denominator_ne_zero hc hyy) |
| have hIU : interior modelSquare ⊆ U := interior_subset.trans hSU |
| have hgI : SmoothPositiveOn g (interior modelSquare) := |
| ⟨fun a b => (hg.1 a b).mono hIU, fun p hp => hg.2 p (hIU hp)⟩ |
| have hBI := (hessianQuotient_bound_from_metric_height_jets hgI isOpen_interior (hz.mono hIU) |
| hG hZ hd hc (fun a b => (hgB a b).restrict_domain interior_subset) |
| (hzB.restrict_domain interior_subset) (fun p hp => hdet p (interior_subset hp)) |
| (fun p hp => hyy p (interior_subset hp))).2 |
| exact ⟨V, hV, hSV, hVU, hVne, hqV, |
| CoordinateBound.of_modelSquare_interior hqV hV hSV hBI⟩ |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
