openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Obstructions /UniversalPatchSelection.lean
| 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 | |