File size: 3,013 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 | import OAI.Analysis.CircleDomains.Reflection.AnnularCircleReflection
namespace OAI
/-! The explicit annular reflection preserves injectivity and keeps
uniform bounds from the exterior side. -/
open Set Function Filter Metric Complex
open scoped Topology ComplexConjugate
noncomputable section
namespace CircleDomainRigidity
lemma annularCircleReflect_outer_iff {U:Set ℂ} {f:ℂ→ℂ}
(hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U)
(hp:∀z∈U,1<‖z‖→1<‖f z‖)
(hb:∀z∈U,‖z‖=1→‖f z‖=1) {z:ℂ} (hzU:z∈U) :
1≤‖annularCircleReflect f z‖ ↔ 1≤‖z‖ := by
by_cases hzo : 1≤‖z‖
· refine iff_of_true ?_ hzo
rw [annularCircleReflect,ite_eq_left hzo]
rcases eq_or_lt_of_le hzo with he|he
· exact (hb z hzU he.symm).ge
· exact (hp z hzU he).le
· have hsmall : ‖z‖<1 := lt_of_not_ge hzo
have href := unitCircleReflection_strict_outer (hz z hzU) hsmall
have hf := hp _ (hs z hzU) href
have hlt : ‖annularCircleReflect f z‖<1 := by
rw [annularCircleReflect,ite_eq_right hzo,unitCircleReflection_norm]
exact (inv_lt_one₀ (zero_lt_one.trans hf)).mpr hf
exact iff_of_false (not_le_of_gt hlt) hzo
lemma injOn_annularCircleReflect {U:Set ℂ} {f:ℂ→ℂ}
(hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U)
(hi:InjOn f (U∩{z | 1≤‖z‖}))
(hp:∀z∈U,1<‖z‖→1<‖f z‖)
(hb:∀z∈U,‖z‖=1→‖f z‖=1) : InjOn (annularCircleReflect f) U := by
intro z hzU w hwU he
have hside : (1≤‖z‖) ↔ 1≤‖w‖ := by
rw [←annularCircleReflect_outer_iff hz hs hp hb hzU,
←annularCircleReflect_outer_iff hz hs hp hb hwU,he]
by_cases hzo : 1≤‖z‖
· have hwo := hside.mp hzo
rw [annularCircleReflect,ite_eq_left hzo,annularCircleReflect,ite_eq_left hwo] at he
exact hi ⟨hzU,hzo⟩ ⟨hwU,hwo⟩ he
· have hwo : ¬1≤‖w‖ := fun h => hzo (hside.mpr h)
rw [annularCircleReflect,ite_eq_right hzo,annularCircleReflect,ite_eq_right hwo] at he
have he' := unitCircleReflection_involutive.injective he
have he'' := hi ⟨hs z hzU,unitCircleReflection_outer (hz z hzU) (lt_of_not_ge hzo).le⟩
⟨hs w hwU,unitCircleReflection_outer (hz w hwU) (lt_of_not_ge hwo).le⟩ he'
exact unitCircleReflection_involutive.injective he''
lemma annularCircleReflect_norm_le {U:Set ℂ} {f:ℂ→ℂ} {M:ℝ}
(hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) (hM:1≤M)
(hf:∀z∈U,1≤‖z‖→1≤‖f z‖ ∧ ‖f z‖≤M) {z:ℂ} (hzU:z∈U) :
‖annularCircleReflect f z‖≤M := by
by_cases hzo : 1≤‖z‖
· rw [annularCircleReflect,ite_eq_left hzo]
exact (hf z hzU hzo).2
· rw [annularCircleReflect,ite_eq_right hzo,unitCircleReflection_norm]
have hh := (hf _ (hs z hzU)
(unitCircleReflection_outer (hz z hzU) (lt_of_not_ge hzo).le)).1
exact ((inv_le_one₀ (zero_lt_one.trans_le hh)).mpr hh).trans hM
end CircleDomainRigidity
end
end OAI
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