File size: 3,584 Bytes
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 | import OAI.Geometry.IsometricImmersion.Obstructions.BoundedClassNowhereDense
import OAI.Geometry.IsometricImmersion.Obstructions.PatchObstructionSet
import OAI.Geometry.IsometricImmersion.Metrics.MetricPatchOpen
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.Model
theorem patchObstructionSet_dense
{g0 : MetricField} {V : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
(hkappa : 0 < kappa)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p) :
Dense (patchObstructionSet g0 kappa) := by
let : BaireSpace (metricPatchSet g0 kappa) := metricPatchSet_baire hg0 hV hSV kappa
change Dense (⋂ index : ℕ × ℚ, (closure (boundedHeightClass g0 kappa index.1 index.2))ᶜ)
apply dense_iInter_of_isOpen (fun _ => isClosed_closure.isOpen_compl)
intro index
exact (isClosed_isNowhereDense_iff_compl.mp ⟨isClosed_closure,
(boundedHeightClass_isNowhereDense hg0 hV hSV hkappa hbackground index.1 index.2).closure⟩).2
theorem no_admissible_height_residual
{g0 : MetricField} {V : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
(hkappa : 0 < kappa)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p) :
{eta : metricPatchSet g0 kappa |
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z} ∈
residual (metricPatchSet g0 kappa) := by
let : BaireSpace (metricPatchSet g0 kappa) := metricPatchSet_baire hg0 hV hSV kappa
apply mem_residual.mpr
refine ⟨patchObstructionSet g0 kappa,?_,patchObstructionSet_isGdelta g0 kappa,
patchObstructionSet_dense hg0 hV hSV hkappa hbackground⟩
intro eta heta
exact patchObstructionSet_has_no_admissible_height heta
theorem exists_patch_obstruction_in_open
{g0 : MetricField} {V : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
(hkappa : 0 < kappa)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p)
{O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O) (hne : O.Nonempty) :
∃ eta ∈ O,
SmoothPositiveOn (perturbedMetric g0 eta.val) V ∧
(∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 eta.val) p < -kappa/2) ∧
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z := by
obtain ⟨eta,hetaGood,hetaO⟩ :=
(patchObstructionSet_dense hg0 hV hSV hkappa hbackground).exists_mem_open hO hne
exact ⟨eta,hetaO,perturbedMetric_smoothPositiveOn hg0 eta.val eta.property.1,
eta.property.2,patchObstructionSet_has_no_admissible_height hetaGood⟩
theorem exists_patch_obstruction_in_neighborhood
{g0 : MetricField} {V : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
(hkappa : 0 < kappa)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p)
{etaCenter : metricPatchSet g0 kappa} {O : Set (metricPatchSet g0 kappa)}
(hO : O ∈ 𝓝 etaCenter) :
∃ eta ∈ O, ∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta.val) z := by
obtain ⟨W,hWO,hW,hcenter⟩ := mem_nhds_iff.mp hO
obtain ⟨eta,hetaW,_,_,hno⟩ :=
exists_patch_obstruction_in_open hg0 hV hSV hkappa hbackground hW ⟨etaCenter,hcenter⟩
exact ⟨eta,hWO hetaW,hno⟩
end SmoothLocal.Perturbation
end
end OAI
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