| import OAI.Analysis.CircleDomains.Uniformization.ConformalComposition |
| import OAI.Analysis.CircleDomains.Uniformization.FiniteCompletion |
|
|
| namespace OAI |
|
|
| /-! |
| # The published finite circle-uniformization input |
|
|
| Conway, *Functions of One Complex Variable II*, GTM 159 (1995), |
| Theorem 15.7.9, p. 106, gives circle uniformization of nondegenerate |
| finitely connected regions. `FiniteCircleUniformization` records only |
| its existence conclusion on the sphere, omitting normalization and |
| uniqueness. It is an explicit proposition parameter, not an axiom. |
|
|
| The complement components of a finite family of disjoint closed |
| connected sets are identified below. This checks the finiteness and |
| nondegeneracy hypotheses for completed surfaces, also after an arbitrary |
| sphere homeomorphism such as compact metric uniformization. |
| -/ |
|
|
| open Set Function |
| open scoped Topology OnePoint |
|
|
| noncomputable section |
|
|
| namespace CircleDomainRigidity |
|
|
| def FiniteNondegenerateComplement (U : Set Sphere) : Prop := |
| (connectedComponentIn Uᶜ '' Uᶜ).Finite ∧ |
| ∀ p ∈ Uᶜ, (connectedComponentIn Uᶜ p).Nontrivial |
|
|
| def FiniteCircleUniformization : Prop := |
| ∀ U : Set Sphere, IsOpen U → IsConnected U → Uᶜ.Nonempty → |
| FiniteNondegenerateComplement U → |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), |
| IsCircleDomain V ∧ IsConformalEquivalence f U V |
|
|
| theorem connectedComponentIn_finite_disjoint_closed |
| {X I : Type*} [TopologicalSpace X] [Finite I] |
| (C : I → Set X) (hC : ∀ i, IsClosed (C i)) |
| (hconn : ∀ i, IsPreconnected (C i)) (hdisj : Pairwise (Disjoint on C)) |
| {i : I} {p : X} (hp : p ∈ C i) : |
| connectedComponentIn (⋃ j, C j) p = C i := by |
| classical |
| let R : Set X := ⋃ j : {j : I // j ≠ i}, C j |
| have hR : IsClosed R := isClosed_iUnion_of_finite (fun j => hC j) |
| have hCR : Disjoint (C i) R := by |
| apply Set.disjoint_left.mpr |
| intro x hx hxR |
| obtain ⟨j, hj⟩ := mem_iUnion.mp hxR |
| exact Set.disjoint_left.mp (hdisj j.property.symm) hx hj |
| have hcover : (⋃ j, C j) ⊆ C i ∪ R := by |
| intro x hx |
| obtain ⟨j, hj⟩ := mem_iUnion.mp hx |
| by_cases hji : j = i |
| · exact Or.inl (hji ▸ hj) |
| · exact Or.inr (mem_iUnion.mpr ⟨⟨j, hji⟩, hj⟩) |
| have hsub : connectedComponentIn (⋃ j, C j) p ⊆ C i := by |
| have hs := isPreconnected_iff_subset_of_disjoint_closed.mp |
| (isPreconnected_connectedComponentIn (F := ⋃ j, C j) (x := p)) |
| (C i) R (hC i) hR ((connectedComponentIn_subset _ _).trans hcover) |
| (by rw [hCR.inter_eq, inter_empty]) |
| rcases hs with h | h |
| · exact h |
| · exact fun x hx => (Set.disjoint_left.mp hCR hp |
| (h (mem_connectedComponentIn (mem_iUnion.mpr ⟨i, hp⟩)))).elim |
| exact hsub.antisymm ((hconn i).subset_connectedComponentIn hp (subset_iUnion C i)) |
|
|
| theorem finite_nondegenerate_complement_of_disjoint_closed |
| {I : Type*} [Finite I] {U : Set Sphere} |
| (C : I → Set Sphere) (hC : ∀ i, IsClosed (C i)) |
| (hconn : ∀ i, IsPreconnected (C i)) (hdisj : Pairwise (Disjoint on C)) |
| (hneq : ∀ i, (C i).Nontrivial) (hcomp : Uᶜ = ⋃ i, C i) : |
| FiniteNondegenerateComplement U := by |
| have hcomponent {p : Sphere} (hp : p ∈ Uᶜ) : |
| ∃ i, connectedComponentIn Uᶜ p = C i := by |
| rw [hcomp] at hp ⊢ |
| obtain ⟨i, hi⟩ := mem_iUnion.mp hp |
| exact ⟨i, connectedComponentIn_finite_disjoint_closed C hC hconn hdisj hi⟩ |
| refine ⟨(finite_range C).subset ?_, ?_⟩ |
| · rintro B ⟨p, hp, rfl⟩ |
| obtain ⟨i, hi⟩ := hcomponent hp |
| exact ⟨i, hi.symm⟩ |
| · intro p hp |
| obtain ⟨i, hi⟩ := hcomponent hp |
| simpa only [hi] using hneq i |
|
|
| theorem circle_model_of_finite_disjoint_complement |
| (hfinite : FiniteCircleUniformization) |
| {I : Type*} [Finite I] {U : Set Sphere} (hU : IsOpen U) (hconnU : IsConnected U) |
| (C : I → Set Sphere) (hC : ∀ i, IsClosed (C i)) |
| (hconn : ∀ i, IsPreconnected (C i)) (hdisj : Pairwise (Disjoint on C)) |
| (hneq : ∀ i, (C i).Nontrivial) (hcomp : Uᶜ = ⋃ i, C i) : |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), |
| IsCircleDomain V ∧ IsConformalEquivalence f U V := by |
| by_cases hne : Uᶜ.Nonempty |
| · exact hfinite U hU hconnU hne |
| (finite_nondegenerate_complement_of_disjoint_closed C hC hconn hdisj hneq hcomp) |
| · have hUniv : U = univ := by |
| apply eq_univ_of_forall |
| intro x |
| by_contra hx |
| exact hne ⟨x, hx⟩ |
| subst U |
| exact circleDomainModel_univ |
|
|
| /-- After compact metric uniformization, the finite circle theorem gives |
| an actual bijection to a circle domain. It is conformal in every region |
| where the metric uniformization retained the original complex structure. -/ |
| theorem circle_model_after_homeomorphism |
| (hfinite : FiniteCircleUniformization) |
| {I : Type*} [Finite I] {U W : Set Sphere} (hU : IsOpen U) (hconnU : IsConnected U) |
| (C : I → Set Sphere) (hC : ∀ i, IsClosed (C i)) |
| (hconn : ∀ i, IsPreconnected (C i)) (hdisj : Pairwise (Disjoint on C)) |
| (hneq : ∀ i, (C i).Nontrivial) (hcomp : Uᶜ = ⋃ i, C i) |
| (e : Sphere ≃ₜ Sphere) (hW : W ⊆ U) (he : IsConformalOn e W) : |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), |
| IsCircleDomain V ∧ BijOn f U V ∧ IsConformalOn f W ∧ |
| IsConformalEquivalence (f ∘ e.symm) (e '' U) V := by |
| have hcomp' : (e '' U)ᶜ = ⋃ i, e '' C i := by |
| rw [← e.image_compl, hcomp, image_iUnion] |
| have hdisj' : Pairwise (Disjoint on fun i => e '' C i) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| rintro y ⟨x, hx, rfl⟩ ⟨z, hz, heq⟩ |
| have hzx : z = x := e.injective heq |
| exact Set.disjoint_left.mp (hdisj hij) hx (hzx ▸ hz) |
| obtain ⟨V, g, hV, hg⟩ := circle_model_of_finite_disjoint_complement hfinite |
| (e.isOpenMap _ hU) (hconnU.image e e.continuous.continuousOn) |
| (fun i => e '' C i) (fun i => e.isClosedMap _ (hC i)) |
| (fun i => (hconn i).image e e.continuous.continuousOn) hdisj' |
| (fun i => (hneq i).image e.injective) hcomp' |
| refine ⟨V, g ∘ e, hV, hg.bijOn.comp e.injective.bijOn_image, |
| hg.1.comp he (fun p hp => ⟨p, hW hp, rfl⟩), ?_⟩ |
| have heq : (g ∘ e) ∘ e.symm = g := by funext p; simp only [Function.comp_apply, e.apply_symm_apply] |
| simpa only [heq] using hg |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
