| 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 |
|
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| end OAI |
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