| import OAI.Analysis.CircleDomains.Topology.ComponentControl |
| import OAI.Analysis.CircleDomains.Modulus.BarrierCollapse |
|
|
| namespace OAI |
|
|
| /-! |
| # From analytic barriers to circle-domain component control |
|
|
| The complementary components are the components of the sphere's closed |
| complement. Components admitting vanishing absolutely continuous barriers |
| are points. If all remaining components are round disks, the domain is |
| a circle domain. |
| -/ |
|
|
| open Set Topology |
| open scoped OnePoint |
|
|
| namespace CircleDomainRigidity |
|
|
| / |
| def planarPart (K : Set Sphere) : Set ℂ := (fun z : ℂ => (z : Sphere)) ⁻¹' K |
|
|
| private theorem image_planarPart {K : Set Sphere} (hinfty : ∞ ∉ K) : |
| (fun z : ℂ => (z : Sphere)) '' planarPart K = K := by |
| apply image_preimage_eq_of_subset |
| intro p hp |
| exact OnePoint.ne_infty_iff_exists.mp (fun h => hinfty (h ▸ hp)) |
|
|
| / |
| plane set in the standard chart. -/ |
| theorem planarPart_isCompact_isConnected {K : Set Sphere} |
| (hcompact : IsCompact K) (hconn : IsConnected K) (hinfty : ∞ ∉ K) : |
| IsCompact (planarPart K) ∧ IsConnected (planarPart K) := by |
| have heq := image_planarPart hinfty |
| have hcomp : IsCompact (planarPart K) := |
| OnePoint.isOpenEmbedding_coe.isEmbedding.isCompact_iff.mpr (heq.symm ▸ hcompact) |
| have hpre : IsPreconnected (planarPart K) := |
| OnePoint.isOpenEmbedding_coe.isEmbedding.isInducing.isPreconnected_image.mp |
| (heq.symm ▸ hconn.isPreconnected) |
| have hne : (planarPart K).Nonempty := by |
| obtain ⟨p, hp⟩ := hconn.nonempty |
| obtain ⟨z, rfl⟩ := OnePoint.ne_infty_iff_exists.mp (fun h => hinfty (h ▸ hp)) |
| exact ⟨z, hp⟩ |
| exact ⟨hcomp, hne, hpre⟩ |
|
|
| / |
| theorem IsComplementaryComponent.isCompact_isConnected {G K : Set Sphere} |
| (hK : IsComplementaryComponent G K) (hG : IsOpen G) : |
| IsCompact K ∧ IsConnected K := by |
| obtain ⟨p, hp, rfl⟩ := hK |
| have hclosed : IsClosed (connectedComponentIn Gᶜ p) := by |
| rw [connectedComponentIn_eq_image hp] |
| exact hG.isClosed_compl.isClosedMap_subtype_val _ isClosed_connectedComponent |
| exact ⟨hclosed.isCompact, isConnected_connectedComponentIn_iff.mpr hp⟩ |
|
|
| / |
| to be a singleton, including components with no interior. -/ |
| theorem IsComplementaryComponent.eq_singleton_of_barriers |
| {G K : Set Sphere} (hK : IsComplementaryComponent G K) |
| (hG : IsOpen G) (hinfty : ∞ ∈ G) |
| (hre : HasVanishingRectangleBarriers |
| (sInf (Complex.re '' planarPart K)) (sSup (Complex.re '' planarPart K))) |
| (him : HasVanishingRectangleBarriers |
| (sInf (Complex.im '' planarPart K)) (sSup (Complex.im '' planarPart K))) : |
| ∃ p : Sphere, K = {p} := by |
| have hKsub : K ⊆ Gᶜ := by |
| obtain ⟨p, _, rfl⟩ := hK |
| exact connectedComponentIn_subset _ _ |
| have hinftyK : ∞ ∉ K := fun h => hKsub h hinfty |
| obtain ⟨hcompact, hconn⟩ := hK.isCompact_isConnected hG |
| obtain ⟨hcompactP, hconnP⟩ := planarPart_isCompact_isConnected hcompact hconn hinftyK |
| have hss := subsingleton_of_vanishing_projection_barriers hcompactP hconnP hre him |
| obtain ⟨z, hz⟩ := hconnP.nonempty |
| have hsingleton : planarPart K = {z} := hss.eq_singleton_of_mem hz |
| refine ⟨(z : Sphere), ?_⟩ |
| rw [← image_planarPart hinftyK, hsingleton] |
| simp |
|
|
| / |
| barrier data. This remains a reduction: barrier existence and round |
| filling have not been assumed to be published results. -/ |
| theorem isCircleDomain_of_component_barriers |
| {G : Set Sphere} (hG : IsOpen G) (hconn : IsConnected G) (hinfty : ∞ ∈ G) |
| (hcontrol : ∀ K, IsComplementaryComponent G K → |
| IsRoundClosedDisk K ∨ |
| (HasVanishingRectangleBarriers |
| (sInf (Complex.re '' planarPart K)) (sSup (Complex.re '' planarPart K)) ∧ |
| HasVanishingRectangleBarriers |
| (sInf (Complex.im '' planarPart K)) (sSup (Complex.im '' planarPart K)))) : |
| IsCircleDomain G := by |
| refine ⟨hG, hconn, ?_⟩ |
| intro p hp |
| have hK : IsComplementaryComponent G (connectedComponentIn Gᶜ p) := ⟨p, hp, rfl⟩ |
| rcases hcontrol _ hK with hdisk | ⟨hre, him⟩ |
| · exact Or.inr hdisk |
| · exact Or.inl (hK.eq_singleton_of_barriers hG hinfty hre him) |
|
|
| end CircleDomainRigidity |
|
|
| end OAI |
|
|
