File size: 4,465 Bytes
9655094 | 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 | 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
/-- The finite-coordinate part of a sphere set. -/
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))
/-- A compact connected sphere set missing infinity is a compact connected
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⟩
/-- A genuine complementary component is compact and connected. -/
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⟩
/-- Vanishing projection barriers force an actual complementary component
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
/-- The final component-recognition implication with explicit analytic
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
|