| import OAI.Analysis.CircleDomains.Reflection.UnitCircleReflection |
|
|
| namespace OAI |
|
|
| /-! Schwarz reflection in a circle on an invariant annular neighborhood. |
| The reflected map is explicit, continuous and holomorphic. -/ |
| open Set Function Filter Metric Complex |
| open scoped Topology ComplexConjugate |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| def annularCircleReflect (f:ℂ→ℂ) (z:ℂ) : ℂ := |
| if 1≤‖z‖ then f z else unitCircleReflection (f (unitCircleReflection z)) |
|
|
| lemma continuousOn_annularCircleReflect {U:Set ℂ} {f:ℂ→ℂ} |
| (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) |
| (hc:ContinuousOn f (U∩{z | 1≤‖z‖})) |
| (hn:∀z∈U,1≤‖z‖→f z≠0) |
| (hb:∀z∈U,‖z‖=1→‖f z‖=1) : |
| ContinuousOn (annularCircleReflect f) U := by |
| let L := U∩{z:ℂ | ‖z‖≤1} |
| have hg : ContinuousOn (fun z => f (unitCircleReflection z)) L := by |
| apply hc.comp ((continuousOn_unitCircleReflection hz).mono inter_subset_left) |
| intro z hzL |
| exact ⟨hs z hzL.1,unitCircleReflection_outer (hz z hzL.1) hzL.2⟩ |
| have hg0 : ∀z∈L,f (unitCircleReflection z)≠0 := fun z hzL => |
| hn _ (hs z hzL.1) (unitCircleReflection_outer (hz z hzL.1) hzL.2) |
| have hgc : ContinuousOn (fun z => unitCircleReflection (f (unitCircleReflection z))) L := |
| (continuous_conj.comp_continuousOn hg).inv₀ (fun z hzL => by simpa using hg0 z hzL) |
| rw [continuousOn_iff_continuous_domRestrict] |
| apply continuous_if_le continuous_const (continuous_norm.comp continuous_subtype_val) |
| · exact hc.comp continuous_subtype_val.continuousOn (fun z hz' => ⟨z.property,hz'⟩) |
| · exact hgc.comp continuous_subtype_val.continuousOn (fun z hz' => ⟨z.property,hz'⟩) |
| · intro z hz' |
| have hzu : ‖(z:ℂ)‖=1 := hz'.symm |
| rw [unitCircleReflection_fixed hzu,unitCircleReflection_fixed (hb z z.property hzu)] |
|
|
| lemma differentiableOn_annularCircleReflect {U:Set ℂ} {f:ℂ→ℂ} |
| (hU:IsOpen U) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) |
| (hc:ContinuousOn f (U∩{z | 1≤‖z‖})) |
| (hd:∀z∈U,1<‖z‖→DifferentiableAt ℂ f z) |
| (hn:∀z∈U,1≤‖z‖→f z≠0) |
| (hb:∀z∈U,‖z‖=1→‖f z‖=1) : |
| DifferentiableOn ℂ (annularCircleReflect f) U := by |
| apply differentiableOn_of_continuousOn_off_unit_circle hU |
| (continuousOn_annularCircleReflect hz hs hc hn hb) |
| intro z hzU hzne |
| rcases lt_or_gt_of_ne hzne with hsmall|hlarge |
| · have he : annularCircleReflect f =ᶠ[𝓝 z] |
| fun w => unitCircleReflection (f (unitCircleReflection w)) := by |
| filter_upwards [(isOpen_lt continuous_norm continuous_const).mem_nhds hsmall] with w hw |
| exact ite_eq_right (not_le_of_gt hw) |
| apply he.differentiableAt_iff.mpr |
| have href := unitCircleReflection_strict_outer (hz z hzU) hsmall |
| have hfd : DifferentiableAt ℂ (fun w:ℂ => f w⁻¹) (conj z) := |
| (hd _ (hs z hzU) href).comp (conj z) |
| (differentiableAt_id.inv (by simpa using hz z hzU)) |
| have hconj : DifferentiableAt ℂ (fun w => conj (f ((conj w)⁻¹))) z := |
| differentiableAt_conj_conj_iff.mpr hfd |
| have hnonzero := hn _ (hs z hzU) href.le |
| exact hconj.inv (by simpa only [unitCircleReflection,ne_eq,map_eq_zero] using hnonzero) |
| · have he : annularCircleReflect f =ᶠ[𝓝 z] f := by |
| filter_upwards [(isOpen_lt continuous_const continuous_norm).mem_nhds hlarge] with w hw |
| exact ite_eq_left hw.le |
| exact he.differentiableAt_iff.mpr (hd z hzU hlarge) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
