File size: 3,214 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 | import OAI.Analysis.CircleDomains.Reflection.SchwarzReflectionInteriorInjectivity
import OAI.Analysis.CircleDomains.Reflection.CayleyHolomorphic
namespace OAI
/-! The circle continuation is injective from interior injectivity alone.
No boundary injectivity or nonzero boundary derivative is assumed. -/
open Set Function Metric Filter Complex
open scoped Topology ComplexConjugate
noncomputable section
namespace CircleDomainRigidity
lemma exists_local_injective_circle_reflection {U : Set ℂ} {f : ℂ → ℂ}
(hU : IsOpen U) (hs : ∀z∈U,conj z∈U)
(hc : ContinuousOn f (U∩{z : ℂ | 0≤z.im}))
(hd : ∀z∈U,0<z.im → DifferentiableAt ℂ f z)
(hi : InjOn f (U∩{z : ℂ | 0<z.im}))
(hinside : ∀z∈U,0<z.im → ‖f z‖<1)
(hcircle : ∀z∈U,z.im=0 → ‖f z‖=1)
(hpole : ∀z∈U,0≤z.im → f z≠1)
{a : ℂ} (ha : a∈U) (ha0 : a.im=0) :
∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ a∈V ∧ V⊆U ∧
DifferentiableOn ℂ F V ∧ InjOn F V ∧ EqOn F f (V∩{z : ℂ | 0≤z.im}) := by
let g : ℂ → ℂ := fun z => diskHalfPlane (f z)
have hgc : ContinuousOn g (U∩{z : ℂ | 0≤z.im}) :=
continuousOn_diskHalfPlane.comp hc (fun z hz => hpole z hz.1 hz.2)
have hgd : ∀z∈U,0<z.im → DifferentiableAt ℂ g z := by
intro z hz hz0
exact (differentiableAt_diskHalfPlane (hpole z hz hz0.le)).comp z (hd z hz hz0)
have hgr : ∀z∈U,z.im=0 → (g z).im=0 := by
intro z hz hz0
exact diskHalfPlane_real (hpole z hz hz0.ge) (hcircle z hz hz0)
have hgi : InjOn g (U∩{z : ℂ | 0<z.im}) := by
intro z hz w hw he
apply hi hz hw
exact diskHalfPlane_injOn_nonpole (hpole z hz.1 (show 0<z.im from hz.2).le)
(hpole w hw.1 (show 0<w.im from hw.2).le) he
have hgp : ∀z∈U,0<z.im → 0<(g z).im :=
fun z hz hz0 => diskHalfPlane_pos (hinside z hz hz0)
let H := schwarzReflect g
have hHi : InjOn H U := injOn_schwarzReflect_of_injOn_upper hU hs hgc hgd hgi hgr hgp
have hHc : ContinuousOn H U := continuousOn_schwarzReflect hs hgc hgr
have hHd : DifferentiableOn ℂ H U := differentiableOn_schwarzReflect hU hs hgc hgd hgr
have hHa : H a+I≠0 := by
change schwarzReflect g a+I≠0
rw [schwarzReflect_eq ha0.ge]
exact diskHalfPlane_add_I_ne_zero (hpole a ha ha0.ge)
have hnbhd : U∩{z : ℂ | H z+I≠0} ∈ 𝓝 a := by
apply inter_mem (hU.mem_nhds ha)
exact ((hHc.continuousAt (hU.mem_nhds ha)).add_const I).eventually
(isOpen_ne |>.mem_nhds hHa)
obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hnbhd
refine ⟨ball a ε,fun z => halfPlaneDisk (H z),isOpen_ball,mem_ball_self hε,
fun z hz => (hball hz).1,?_,?_,?_⟩
· intro z hz
exact ((differentiableAt_halfPlaneDisk (hball hz).2).comp z
((hHd z (hball hz).1).differentiableAt (hU.mem_nhds (hball hz).1))).differentiableWithinAt
· intro z hz w hw he
apply hHi (hball hz).1 (hball hw).1
exact halfPlaneDisk_injOn_nonpole (hball hz).2 (hball hw).2 he
· intro z hz
change halfPlaneDisk (schwarzReflect g z)=f z
rw [schwarzReflect_eq hz.2]
exact halfPlaneDisk_diskHalfPlane (hpole z (hball hz.1).1 hz.2)
end CircleDomainRigidity
end
end OAI
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