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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 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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import OAI.Geometry.IsometricImmersion.Immersions.OrientedHeightPullback
import OAI.Geometry.IsometricImmersion.Obstructions.DiskBoundaryObstruction
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators Matrix
namespace SmoothLocal.Geometry
theorem rank_one_boundary_eventually_admissible_patches
{g : MetricField} {F : Coord → Ambient} {U : Set Coord} {b : Coord}
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U)
(hbU : b ∈ U) (n : ℕ) (hb : b ∈ frontier (accumulatingDisk n))
(e : Ambient) (he : IsUnitNormalAt F e b)
(hrank : (secondFundamental F e b).rank = 1) :
∃ R : OrientationLabel, ∃ centers : ℕ → Coord,
(∀ k, centers k ∈ orientedExteriorCenters n k R) ∧
Tendsto centers atTop (𝓝 b) ∧
∀ᶠ k in atTop,
orientedClosedPatch n k R (centers k) ⊆ U ∧
PatchAdmissibleHeight
(affinePullbackMetric g (centers k) (orientedPatchScale n k R • R.val))
(height (affinePullbackImmersion F (centers k)
(orientedPatchScale n k R • R.val)) e) := by
obtain ⟨R, V, hVo, hbV, hVU, hgood⟩ :=
rank_one_height_has_oriented_neighborhood hg hF hU hbU e he hrank
obtain ⟨centers, hcenters, ht, hpatches⟩ :=
every_boundary_point_every_orientation_patch_neighborhoods n b hb R
refine ⟨R, centers, hcenters, ht, ?_⟩
filter_upwards [hpatches V (hVo.mem_nhds hbV)] with k hk
exact ⟨hk.trans hVU,
oriented_patch_height_admissible_of_subset hg hF hU e he.1 hVU R hgood
n k (centers k) hk⟩
theorem rank_one_boundary_exists_admissible_patch
{g : MetricField} {F : Coord → Ambient} {U : Set Coord} {b : Coord}
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U)
(hbU : b ∈ U) (n : ℕ) (hb : b ∈ frontier (accumulatingDisk n))
(e : Ambient) (he : IsUnitNormalAt F e b)
(hrank : (secondFundamental F e b).rank = 1) (kmin : ℕ) :
∃ R : OrientationLabel, ∃ k ≥ kmin, ∃ c ∈ orientedExteriorCenters n k R,
orientedClosedPatch n k R c ⊆ U ∧
PatchAdmissibleHeight
(affinePullbackMetric g c (orientedPatchScale n k R • R.val))
(height (affinePullbackImmersion F c (orientedPatchScale n k R • R.val)) e) := by
obtain ⟨R, centers, hcenters, _, hgood⟩ :=
rank_one_boundary_eventually_admissible_patches hg hF hU hbU n hb e he hrank
obtain ⟨k, hk⟩ := (hgood.and (eventually_ge_atTop kmin)).exists
exact ⟨R, k, hk.2, centers k, hcenters k, hk.1.1, hk.1.2⟩
theorem immersion_containing_disk_has_admissible_patch
{g : MetricField} {F : Coord → Ambient} {U : Set Coord}
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U)
(n : ℕ) (hDU : accumulatingDisk n ⊆ U)
(hK : ∀ p ∈ roundOpenDisk (accumulatingCenter n) (accumulatingRadius n),
gaussianCurvature g p < 0)
(hKb : ∀ p ∈ frontier (accumulatingDisk n), gaussianCurvature g p = 0)
(kmin : ℕ) :
∃ b ∈ frontier (accumulatingDisk n), ∃ e : Ambient,
IsUnitNormalAt F e b ∧ secondFundamental F e b ≠ 0 ∧
(secondFundamental F e b).rank = 1 ∧
∃ R : OrientationLabel, ∃ k ≥ kmin, ∃ c ∈ orientedExteriorCenters n k R,
orientedClosedPatch n k R c ⊆ U ∧
PatchAdmissibleHeight
(affinePullbackMetric g c (orientedPatchScale n k R • R.val))
(height (affinePullbackImmersion F c (orientedPatchScale n k R • R.val)) e) := by
obtain ⟨b, hb, e, he, hne, hrank⟩ :=
exists_roundDisk_boundary_unitNormal_rank_one hg hF hU (accumulatingRadius_pos n)
hDU hK hKb
have hbD : b ∈ accumulatingDisk n :=
((mem_frontier_roundClosedDisk_iff (accumulatingCenter n) b
(accumulatingRadius_pos n)).mp hb).le
refine ⟨b, hb, e, he, hne, hrank, ?_⟩
exact rank_one_boundary_exists_admissible_patch hg hF hU (hDU hbD) n hb e he hrank kmin
theorem every_local_immersion_has_admissible_oriented_patch
{g : MetricField} {F : Coord → Ambient} {U : Set Coord}
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U)
(h0 : (0 : Coord) ∈ U)
(hK : ∀ n : ℕ, ∀ p ∈ roundOpenDisk (accumulatingCenter n) (accumulatingRadius n),
gaussianCurvature g p < 0)
(hKb : ∀ n : ℕ, ∀ p ∈ frontier (accumulatingDisk n), gaussianCurvature g p = 0)
(nmin kmin : ℕ) :
∃ n ≥ nmin, ∃ b ∈ frontier (accumulatingDisk n), ∃ e : Ambient,
IsUnitNormalAt F e b ∧ secondFundamental F e b ≠ 0 ∧
(secondFundamental F e b).rank = 1 ∧
∃ R : OrientationLabel, ∃ k ≥ kmin, ∃ c ∈ orientedExteriorCenters n k R,
orientedClosedPatch n k R c ⊆ U ∧
PatchAdmissibleHeight
(affinePullbackMetric g c (orientedPatchScale n k R • R.val))
(height (affinePullbackImmersion F c (orientedPatchScale n k R • R.val)) e) := by
have hdisks := eventually_accumulatingDisk_subset (hU.mem_nhds h0)
obtain ⟨n, hn⟩ := (hdisks.and (eventually_ge_atTop nmin)).exists
exact ⟨n, hn.2, immersion_containing_disk_has_admissible_patch hg hF hU n hn.1
(hK n) (hKb n) kmin⟩
end SmoothLocal.Geometry
end
end OAI
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