| import OAI.Geometry.Immersion.ClosedSurface.AtlasMargins |
| import OAI.Geometry.Immersion.ClosedSurface.AtlasPartition |
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| namespace OAI |
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| noncomputable section |
| open Set Complex Bundle Manifold |
| open scoped ContDiff Matrix Topology Manifold BigOperators |
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| namespace ClosedSurfaceR4 |
| open Set PhaseGeometry SmallModes RealModes PhaseGrid |
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| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold planeModel ∞ M] [T2Space M] [CompactSpace M] |
| {ι : Type*} [Fintype ι] |
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| abbrev AtlasPhaseLabel := ι × Fin (Fintype.card PhaseGrid.Label) |
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| def atlasPhaseLabel (a : ι × (Index × Fin 3)) : Fin (Fintype.card (AtlasPhaseLabel (ι := ι))) := |
| Fintype.equivFin (AtlasPhaseLabel (ι := ι)) (a.1,phaseLabel a.2) |
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| lemma atlas_phase_label_count : Fintype.card (AtlasPhaseLabel (ι := ι)) = Fintype.card ι * 108 := by |
| simp [AtlasPhaseLabel,PhaseGrid.Label] |
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| omit [IsManifold planeModel ∞ M] [T2Space M] [CompactSpace M] in |
| theorem atlas_distinct_labels_on_intersection (p : ι → M) (ψ : ι → M → ℝ) |
| (hsource : ∀ i, tsupport (ψ i) ⊆ (chartAt Plane (p i)).source) |
| (s : ι → Finset Index) {h : ℝ} (hh : 0 < h) |
| {a b : ι × (Index × Fin 3)} (hab : a ≠ b) {q : M} |
| (ha : q ∈ tsupport (refinedCutoff (p a.1) (ψ a.1) (s a.1) h a.2.1)) |
| (hb : q ∈ tsupport (refinedCutoff (p b.1) (ψ b.1) (s b.1) h b.2.1)) : |
| atlasPhaseLabel a ≠ atlasPhaseLabel b := by |
| intro he |
| have hl := (Fintype.equivFin (AtlasPhaseLabel (ι := ι))).injective he |
| have hi : a.1 = b.1 := congrArg (fun x : AtlasPhaseLabel (ι := ι) => x.1) hl |
| rcases a with ⟨i,a⟩ |
| rcases b with ⟨j,b⟩ |
| simp only at hi |
| subst j |
| have hab' : a ≠ b := fun he => hab (congrArg (fun x => (i,x)) he) |
| have hca := normalizedCutoff_tsupport (s i) h a.1 |
| (refinedCutoff_tsupport_inner (p i) (hsource i) (s i) h a.1 ha) |
| have hcb := normalizedCutoff_tsupport (s i) h b.1 |
| (refinedCutoff_tsupport_inner (p i) (hsource i) (s i) h b.1 hb) |
| exact distinct_labels_on_intersection hh hab' hca hcb (congrArg (fun x : AtlasPhaseLabel (ι := ι) => x.2) hl) |
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| end ClosedSurfaceR4 |
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| namespace ClosedSurfaceR4 |
| open SmallModes RealModes PhaseGeometry Set Filter |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] |
| [IsManifold planeModel ∞ M] {ι : Type*} [Fintype ι] |
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| theorem finite_atlas_transition_bounds (p : ι → M) (ψ : ι → M → ℝ) |
| (hsource : ∀ i, tsupport (ψ i) ⊆ (chartAt Plane (p i)).source) |
| (hcompact : ∀ i, HasCompactSupport (ψ i)) : |
| ∃ d L : ℝ, 0 < d ∧ 0 < L ∧ ∀ i j : ι, |
| ∀ x ∈ coordinateChart (p i) '' (tsupport (ψ i) ∩ tsupport (ψ j)), |
| d ≤ |coordDet (fderiv ℝ (coordinateTransition (p j) (p i)) x)| ∧ |
| ‖fderiv ℝ (coordinateTransition (p j) (p i)) x‖ ≤ L := by |
| classical |
| have hs (i : ι) : tsupport (ψ i) ⊆ (coordinateChart (p i)).source := by |
| rw [coordinateChart_source] |
| exact hsource i |
| have hb (a : ι × ι) : ∃ d L : ℝ, 0 < d ∧ 0 < L ∧ |
| ∀ x ∈ coordinateChart (p a.1) '' (tsupport (ψ a.1) ∩ tsupport (ψ a.2)), |
| d ≤ |coordDet (fderiv ℝ (coordinateTransition (p a.2) (p a.1)) x)| ∧ |
| ‖fderiv ℝ (coordinateTransition (p a.2) (p a.1)) x‖ ≤ L := by |
| have hK := (hcompact a.1).inter_right (isClosed_tsupport (ψ a.2)) |
| exact coordinateTransition_compact_bounds (p a.2) (p a.1) |
| (coordinateChart_compact_image hK (p a.1) (fun x hx => hs a.1 hx.1)) |
| (coordinateChart_image_overlap (p a.2) (p a.1) |
| (fun x hx => hs a.1 hx.1) (fun x hx => hs a.2 hx.2)) |
| choose d L hd hL hbound using hb |
| obtain ⟨d₀,hd₀,_,hle⟩ := finite_positive_threshold d hd |
| refine ⟨d₀,1+∑ a, L a,hd₀,?_,?_⟩ |
| · have hn : 0 ≤ ∑ a, L a := Finset.sum_nonneg (fun a _ => le_of_lt (hL a)) |
| linarith |
| · intro i j x hx |
| refine ⟨(hle (i,j)).trans ((hbound (i,j) x hx).1),?_⟩ |
| have hsL : L (i,j) ≤ ∑ a, L a := |
| Finset.single_le_sum (fun a _ => le_of_lt (hL a)) (Finset.mem_univ (i,j)) |
| exact ((hbound (i,j) x hx).2).trans (by linarith) |
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| end ClosedSurfaceR4 |
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| end |
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| end OAI |
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