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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 41 42 43 | import OAI.Geometry.IsometricImmersion.Pulses.OriginalPulseExclusion
import OAI.Geometry.IsometricImmersion.Obstructions.BoundedClassTopology
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.Flow SmoothLocal.Model
theorem boundedHeightClass_isNowhereDense
{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)
(M : ℕ) (q0 : ℚ) :
IsNowhereDense (boundedHeightClass g0 kappa M q0) := by
change interior (closure (boundedHeightClass g0 kappa M q0)) = ∅
apply Set.eq_empty_iff_forall_notMem.mpr
intro etaStar hStar
let O := interior (closure (boundedHeightClass g0 kappa M q0))
have hO : IsOpen O := isOpen_interior
have hOclosure : O ⊆ closure (boundedHeightClass g0 kappa M q0) := interior_subset
have hparameters := boundedHeightClass_closure_parameters (hOclosure hStar)
have hgStar : SmoothPositiveOn (perturbedMetric g0 etaStar.val) V :=
perturbedMetric_smoothPositiveOn hg0 etaStar.val etaStar.property.1
obtain ⟨r,hr,_,hLr,hexclusion⟩ := exists_original_pulse_exclusion
hg0 hgStar hV hSV M hkappa hparameters.1 hparameters.2.le hbackground
obtain ⟨N,hN,happroximants⟩ := bounded_class_open_closure_pulse_family_at_radius
hg0 hV hSV M q0 hO hStar hOclosure hr hLr
obtain ⟨delta,hdelt,_,hnoheight⟩ := hexclusion N hN
obtain ⟨tau,happroxTau,hnoheightTau⟩ := ((happroximants delta hdelt).and hnoheight).exists
obtain ⟨eta,z,hclass,hcenter,haccuracy⟩ := happroxTau.2
exact hnoheightTau eta z hclass hcenter haccuracy
end SmoothLocal.Perturbation
end
end OAI
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