| import OAI.Analysis.DirectCrouzeix.HolomorphicReflection | |
| namespace OAI | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| noncomputable section | |
| open MeasureTheory Set Filter Metric | |
| open scoped Topology Interval ENNReal NNReal ComplexConjugate | |
| noncomputable section | |
| open Filter Metric Set | |
| open scoped Topology ComplexConjugate | |
| namespace DirectCrouzeix | |
| namespace Conformal | |
| open Function Complex | |
| open scoped Pointwise | |
| open InnerProductSpace Real | |
| theorem boundary_germs_compatible {U : Set ℂ} (hU : IsOpen U) | |
| {p q z : ℂ} (hp : p ∈ closure U) (hq : q ∈ closure U) | |
| {r s : ℝ} (hr : 0 < r) (hs : 0 < s) | |
| (hzp : z ∈ ball p r) (hzq : z ∈ ball q s) | |
| {E G f : ℂ → ℂ} | |
| (hE : AnalyticOnNhd ℂ E (ball p (3*r))) | |
| (hG : AnalyticOnNhd ℂ G (ball q (3*s))) | |
| (he : EqOn E f (ball p (3*r) ∩ U)) | |
| (hg : EqOn G f (ball q (3*s) ∩ U)) : E z = G z := by | |
| let V := ball p (3*r) ∩ ball q (3*s) | |
| have hVo : IsOpen V := isOpen_ball.inter isOpen_ball | |
| have hmeet : (V ∩ closure U).Nonempty := by | |
| have hd : dist p q < r + s := by | |
| have := dist_triangle p z q | |
| rw [dist_comm p z] at this | |
| have hpz := mem_ball.mp hzp | |
| have hqz := mem_ball.mp hzq | |
| linarith | |
| rcases le_total r s with hrs | hsr | |
| · refine ⟨p, ⟨mem_ball_self (by positivity), ?_⟩, hp⟩ | |
| rw [mem_ball] | |
| linarith | |
| · refine ⟨q, ⟨?_, mem_ball_self (by positivity)⟩, hq⟩ | |
| rw [mem_ball, dist_comm] | |
| linarith | |
| obtain ⟨b, hbV, hbU⟩ := hmeet | |
| obtain ⟨a, haV, haU⟩ := mem_closure_iff_nhds.mp hbU V (hVo.mem_nhds hbV) | |
| have heq : E =ᶠ[𝓝 a] G := by | |
| filter_upwards [(hVo.inter hU).mem_nhds ⟨haV, haU⟩] with w hw | |
| exact (he ⟨hw.1.1, hw.2⟩).trans (hg ⟨hw.1.2, hw.2⟩).symm | |
| exact (hE.mono inter_subset_left).eqOn_of_preconnected_of_eventuallyEq | |
| (hG.mono inter_subset_right) | |
| ((convex_ball p (3*r)).inter (convex_ball q (3*s))).isPreconnected haV heq | |
| ⟨ball_subset_ball (by linarith) hzp, ball_subset_ball (by linarith) hzq⟩ | |
| theorem patch_boundary_germs {U : Set ℂ} (hU : IsOpen U) {f : ℂ → ℂ} | |
| (hloc : ∀ p ∈ closure U, ∃ r > 0, ∃ E : ℂ → ℂ, | |
| AnalyticOnNhd ℂ E (ball p (3*r)) ∧ EqOn E f (ball p (3*r) ∩ U) ∧ | |
| (∀ z ∈ ball p (3*r), ‖E z‖ < 1 ↔ z ∈ U) ∧ deriv E p ≠ 0) : | |
| ∃ V : Set ℂ, ∃ F : ℂ → ℂ, IsOpen V ∧ closure U ⊆ V ∧ | |
| AnalyticOnNhd ℂ F V ∧ EqOn F f U ∧ | |
| (∀ z ∈ V, ‖F z‖ < 1 ↔ z ∈ U) ∧ (∀ p ∈ closure U, deriv F p ≠ 0) := by | |
| classical | |
| choose r hr E hE he hside hd using hloc | |
| let V : Set ℂ := ⋃ p : closure U, ball (p : ℂ) (r p p.property) | |
| have hV : IsOpen V := isOpen_iUnion fun _ => isOpen_ball | |
| have hUV : closure U ⊆ V := by | |
| intro p hp | |
| exact mem_iUnion.mpr ⟨⟨p,hp⟩, mem_ball_self (hr p hp)⟩ | |
| have hchoose : ∀ z ∈ V, ∃ p : closure U, z ∈ ball (p : ℂ) (r p p.property) := by | |
| intro z hz | |
| exact mem_iUnion.mp hz | |
| let F : ℂ → ℂ := fun z => if hz : z ∈ V then | |
| E (hchoose z hz).choose (hchoose z hz).choose.property z else f z | |
| have hFE : ∀ p : closure U, EqOn F (E p p.property) (ball (p : ℂ) (r p p.property)) := by | |
| intro p z hz | |
| have hzV : z ∈ V := mem_iUnion.mpr ⟨p,hz⟩ | |
| dsimp only [F] | |
| rw [dite_eq_left hzV] | |
| exact boundary_germs_compatible hU (hchoose z hzV).choose.property p.property | |
| (hr _ _) (hr _ _) (hchoose z hzV).choose_spec hz | |
| (hE _ _) (hE _ _) (he _ _) (he _ _) | |
| refine ⟨V,F,hV,hUV,?_,?_,?_,?_⟩ | |
| · intro z hz | |
| obtain ⟨p,hp⟩ := hchoose z hz | |
| apply (hE p p.property z (ball_subset_ball (by linarith [hr p p.property]) hp)).congr | |
| filter_upwards [isOpen_ball.mem_nhds hp] with w hw | |
| exact (hFE p hw).symm | |
| · intro z hz | |
| exact (hFE ⟨z,subset_closure hz⟩ (mem_ball_self (hr _ _))).trans | |
| (he z (subset_closure hz) ⟨mem_ball_self (by linarith [hr z (subset_closure hz)]),hz⟩) | |
| · intro z hz | |
| obtain ⟨p,hp⟩ := hchoose z hz | |
| rw [hFE p hp] | |
| exact hside p p.property z (ball_subset_ball (by linarith [hr p p.property]) hp) | |
| · intro p hp | |
| have hevent : F =ᶠ[𝓝 p] E p hp := by | |
| filter_upwards [ball_mem_nhds p (hr p hp)] with z hz | |
| exact hFE ⟨p,hp⟩ hz | |
| rw [hevent.deriv_eq] | |
| exact hd p hp | |
| theorem injective_closure_of_disk_side {U V : Set ℂ} (hV : IsOpen V) | |
| (hUV : closure U ⊆ V) {F : ℂ → ℂ} (hF : AnalyticOnNhd ℂ F V) | |
| (hi : InjOn F U) (hm : F '' U = ball 0 1) | |
| (hside : ∀ z ∈ V, ‖F z‖ < 1 ↔ z ∈ U) | |
| (hd : ∀ p ∈ closure U, deriv F p ≠ 0) : InjOn F (closure U) := by | |
| intro p hp q hq he | |
| by_contra hpq | |
| obtain ⟨S,T,hS,hT,hpS,hqT,hST⟩ := t2_separation hpq | |
| have hpN : S ∩ V ∈ 𝓝 p := (hS.inter hV).mem_nhds ⟨hpS,hUV hp⟩ | |
| have hqN : T ∩ V ∈ 𝓝 q := (hT.inter hV).mem_nhds ⟨hqT,hUV hq⟩ | |
| have hpI : F '' (S ∩ V) ∈ 𝓝 (F p) := by | |
| rw [← (hF p (hUV hp)).hasStrictDerivAt.map_nhds_eq (hd p hp)] | |
| exact Filter.image_mem_map hpN | |
| have hqI : F '' (T ∩ V) ∈ 𝓝 (F p) := by | |
| rw [he, ← (hF q (hUV hq)).hasStrictDerivAt.map_nhds_eq (hd q hq)] | |
| exact Filter.image_mem_map hqN | |
| have hpCl : F p ∈ closure (ball (0 : ℂ) 1) := by | |
| rw [← hm] | |
| exact mem_closure_image (hF p (hUV hp)).continuousAt hp | |
| obtain ⟨w,hwI,hwD⟩ := mem_closure_iff_nhds.mp hpCl _ (inter_mem hpI hqI) | |
| obtain ⟨a,haS,haw⟩ := hwI.1 | |
| obtain ⟨b,hbT,hbw⟩ := hwI.2 | |
| have haU : a ∈ U := (hside a haS.2).mp (by rw [haw]; exact mem_ball_zero_iff.mp hwD) | |
| have hbU : b ∈ U := (hside b hbT.2).mp (by rw [hbw]; exact mem_ball_zero_iff.mp hwD) | |
| have hab : a = b := hi haU hbU (haw.trans hbw.symm) | |
| exact Set.disjoint_left.mp hST haS.1 (hab ▸ hbT.1) | |
| theorem univalent_neighborhood_of_disk_side {U V : Set ℂ} (hV : IsOpen V) | |
| (hUV : closure U ⊆ V) (hUc : IsCompact (closure U)) | |
| {F : ℂ → ℂ} (hF : AnalyticOnNhd ℂ F V) | |
| (hi : InjOn F U) (hm : F '' U = ball 0 1) | |
| (hside : ∀ z ∈ V, ‖F z‖ < 1 ↔ z ∈ U) | |
| (hd : ∀ p ∈ closure U, deriv F p ≠ 0) : | |
| ∃ W, IsOpen W ∧ closure U ⊆ W ∧ W ⊆ V ∧ | |
| InjOn F W ∧ AnalyticOnNhd ℂ F W ∧ F '' closure U = closedBall 0 1 := by | |
| have hic := injective_closure_of_disk_side hV hUV hF hi hm hside hd | |
| have hlocal : ∀ p ∈ closure U, ∃ S ∈ 𝓝 p, InjOn F S := by | |
| intro p hp | |
| let d := (hF p (hUV hp)).hasStrictDerivAt.hasStrictFDerivAt_equiv (hd p hp) | |
| exact ⟨(d.toOpenPartialHomeomorph F).source, | |
| (d.toOpenPartialHomeomorph F).open_source.mem_nhds d.mem_toOpenPartialHomeomorph_source, | |
| (d.toOpenPartialHomeomorph F).injOn⟩ | |
| obtain ⟨W,hW,hUW,hiW⟩ := hic.exists_isOpen_superset hUc | |
| (fun p hp => (hF p (hUV hp)).continuousAt) hlocal | |
| refine ⟨W ∩ V,hW.inter hV,fun z hz => ⟨hUW hz,hUV hz⟩,inter_subset_right, | |
| hiW.mono inter_subset_left,hF.mono inter_subset_right,?_⟩ | |
| apply Subset.antisymm | |
| · rintro w ⟨z,hz,rfl⟩ | |
| rw [← closure_ball (0 : ℂ) (by norm_num : (1 : ℝ) ≠ 0), ← hm] | |
| exact mem_closure_image (hF z (hUV hz)).continuousAt hz | |
| · have hc : IsClosed (F '' closure U) := | |
| (hUc.image_of_continuousOn (hF.continuousOn.mono hUV)).isClosed | |
| rw [← closure_ball (0 : ℂ) (by norm_num : (1 : ℝ) ≠ 0)] | |
| apply hc.closure_subset_iff.mpr | |
| rw [← hm] | |
| exact image_mono subset_closure | |
| end Conformal | |
| end DirectCrouzeix | |
| end | |
| end | |
| end | |
| end OAI | |