File size: 3,385 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 | import OAI.Geometry.IsometricImmersion.Obstructions.PatchResidualTheorem
import OAI.Geometry.IsometricImmersion.Metrics.AffineTensorNeighborhood
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.Model
theorem model_zero_mem_metricPatchSet
{g0 : MetricField} {V : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 V) (hSV : modelSquare ⊆ V) (hkappa : 0 < kappa)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p=modelCurvature kappa p) :
(0 : SymmetricPerturbation) ∈ metricPatchSet g0 kappa := by
apply zero_mem_metricPatchSet
· intro p hp
exact hg0.2 p (hSV hp)
· intro p hp
rw [hbackground p (hSV (centralBox_subset_modelSquare hp))]
exact modelCurvature_central_square hkappa (fun i => ⟨hp.1 i,hp.2 i⟩)
theorem exists_supported_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 SymmetricPerturbation} (hO : IsOpen O) (hzero : (0 : SymmetricPerturbation) ∈ O) :
∃ eta : SymmetricPerturbation, eta ∈ O ∧ eta ∈ metricPatchSet g0 kappa ∧
SmoothPositiveOn (perturbedMetric g0 eta) V ∧
∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta) z := by
let zeroPatch : metricPatchSet g0 kappa :=
⟨0,model_zero_mem_metricPatchSet hg0 hSV hkappa hbackground⟩
let W : Set (metricPatchSet g0 kappa) := Subtype.val ⁻¹' O
have hW : IsOpen W := hO.preimage continuous_subtype_val
have hWzero : zeroPatch ∈ W := hzero
obtain ⟨eta,hetaW,hg,_,hno⟩ :=
exists_patch_obstruction_in_open hg0 hV hSV hkappa hbackground hW ⟨zeroPatch,hWzero⟩
exact ⟨eta.val,hetaW,eta.property,hg,hno⟩
theorem exists_small_affine_patch_obstruction
{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)
(b : Coord) (R : Matrix (Fin 2) (Fin 2) ℝ) (N : ℕ)
{epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ eta : SymmetricPerturbation,
eta ∈ metricPatchSet g0 kappa ∧ SmoothPositiveOn (perturbedMetric g0 eta) V ∧
(∀ z : Coord → ℝ, ¬ PatchAdmissibleHeight (perturbedMetric g0 eta) z) ∧
(∀ k ≤ N, ∀ p : Coord,
‖iteratedFDeriv ℝ k (affinePushforwardMetric (perturbationTensor eta) b R) p‖ < epsilon) ∧
∀ i j : Fin 2, ∀ k ≤ N, ∀ p : Coord,
‖iteratedFDeriv ℝ k
(fun q => affinePushforwardMetric (perturbationTensor eta) b R q i j) p‖ < epsilon := by
obtain ⟨eta,hetaO,hetaMetric,hg,hno⟩ := exists_supported_patch_obstruction_in_open
hg0 hV hSV hkappa hbackground (affineTensorJetNeighborhood_isOpen R N epsilon)
(zero_mem_affineTensorJetNeighborhood R N hepsilon)
exact ⟨eta,hetaMetric,hg,hno,
fun k hk p => affineTensorJetNeighborhood_physical_bounds b R N hetaO hk p,
fun i j k hk p => affineTensorJetNeighborhood_coefficient_bounds b R N hetaO hk p i j⟩
end SmoothLocal.Perturbation
end
end OAI
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