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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 | 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
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