File size: 3,099 Bytes
a97875e | 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 | import OAI.Analysis.CircleDomains.Transfer.SourceEndProjection
namespace OAI
/-! Nested matched caps prove continuity of the source quotient projection
at every complementary point, including nondegenerate target components. -/
open Set Function Filter
open scoped Topology
noncomputable section
namespace CircleDomainRigidity.FiniteTransfer
variable {S : Type*} {G : Set Sphere} {π : Sphere → Sphere} {σ : S → Sphere}
{d : CountableRealCoordinates G π σ} {O : ℕ → S → Set Sphere} {ε : ℕ → ℝ}
{M : RealModelExhaustion d O ε}
namespace AssignedRealModelLimit
variable (L : AssignedRealModelLimit M)
theorem sourceProjection_continuous (hG : IsOpen G)
(π : Sphere → Sphere) (hπ : Continuous π)
(hc : ∀ q p, p ∈ connectedComponentIn Gᶜ q → π p = π q) :
Continuous (L.sourceProjection hG π) := by
rw [continuous_iff_continuousAt]
intro p
by_cases hp : p ∈ extendAtInfinity L.function '' G
· obtain ⟨x, hx, rfl⟩ := hp
have hi := continuousAt_conformal_invFunOn L.conformal.1 hG L.conformal.bijOn.injOn hx
have hcont := hπ.continuousAt.comp hi
have hval : L.sourceProjection hG π (extendAtInfinity L.function x) =
π (invFunOn (extendAtInfinity L.function) G (extendAtInfinity L.function x)) := by
simp only [sourceProjection, dite_eq_left (mem_image_of_mem _ hx)]
change Tendsto (L.sourceProjection hG π) (𝓝 (extendAtInfinity L.function x))
(𝓝 (L.sourceProjection hG π (extendAtInfinity L.function x)))
rw [hval]
apply hcont.congr'
have hV := L.conformal.1.isOpen_image hG hG (Subset.refl G)
filter_upwards [hV.mem_nhds (mem_image_of_mem _ hx)] with z hz
simp only [sourceProjection, dite_eq_left hz, comp_apply]
· let A := L.chosenSourceEnd hG p hp
have hval : L.sourceProjection hG π p = π A.source := L.sourceProjection_off_image hG π hp
change Tendsto (L.sourceProjection hG π) (𝓝 p) (𝓝 (L.sourceProjection hG π p))
rw [hval]
apply tendsto_nhds.mpr
intro V hV hpV
have hsub : connectedComponentIn Gᶜ A.source ⊆ π ⁻¹' V := by
intro x hx
change π x ∈ V
rw [hc A.source x hx]
exact hpV
obtain ⟨n, hn⟩ := (A.matching.eventually_source_cap_subset (hV.preimage hπ) hsub).exists
have hpCap : p ∈ ((↑) : ℂ → Sphere) '' jordanInterior (L.contours n (A.matching.address n)) := by
have ht := A.matching.nested n
(mem_iInter.mp (A.matching.target ▸ mem_connectedComponentIn A.matching.outside) (n + 1))
exact (congrArg (fun t : Sphere => t ∈ ((↑) : ℂ → Sphere) ''
jordanInterior (L.contours n (A.matching.address n))) A.point_eq).mp ht
have hOpen := OnePoint.isOpen_image_coe.mpr
(jordanInterior_isOpen (L.contours n (A.matching.address n)))
filter_upwards [hOpen.mem_nhds hpCap] with z hz
obtain ⟨x, hx, hproj⟩ := A.matching.projection_in_source_cap hG π n hz
change L.sourceProjection hG π z ∈ V
rw [hproj]
exact hn hx
end AssignedRealModelLimit
end CircleDomainRigidity.FiniteTransfer
end
end OAI
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