| import OAI.Geometry.IsometricImmersion.Caps.BoundedClassReflection |
| import OAI.Geometry.IsometricImmersion.Caps.CapInductionData |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Perturbation |
| open SmoothLocal.Geometry SmoothLocal.Flow |
|
|
| theorem bounded_class_actual_cap_data |
| {g0 : MetricField} {V : Set Coord} {kappa : ℝ} |
| (hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) |
| (eta : metricPatchSet g0 kappa) {z : Coord → ℝ} {M : ℕ} |
| (hclass : BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z) : |
| ∃ U : Set Coord, IsOpen U ∧ modelSquare ⊆ U ∧ U ⊆ V ∧ |
| SmoothPositiveOn g0 U ∧ SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧ |
| CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z := by |
| obtain ⟨_,hadm,hjets,hyy,hE⟩ := hclass |
| obtain ⟨W,hW,hSW,hg,hz,hD,_,_,hq⟩ := hadm |
| let U := W ∩ V |
| have hU : IsOpen U := hW.inter hV |
| have hSU : modelSquare ⊆ U := fun p hp => ⟨hSW hp,hSV hp⟩ |
| have hg0U : SmoothPositiveOn g0 U := |
| ⟨fun i j => (hg0.1 i j).mono inter_subset_right, |
| fun p hp => hg0.2 p hp.2⟩ |
| have hgU : SmoothPositiveOn (perturbedMetric g0 eta.val) U := |
| ⟨fun i j => (hg.1 i j).mono inter_subset_left, |
| fun p hp => hg.2 p hp.1⟩ |
| refine ⟨U,hU,hSU,inter_subset_right,hg0U,hgU,?_⟩ |
| exact ⟨hz.mono inter_subset_left,hjets,(fun p hp => (hyy p hp).1),hE,hq,hD⟩ |
|
|
| end SmoothLocal.Perturbation |
|
|
| end |
|
|
| end OAI |
|
|
