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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 | import OAI.Analysis.CircleDomains.Reflection.MovingCircleReflection
import OAI.Analysis.CircleDomains.Uniformization.MontelProof
namespace OAI
/-! Compactness of the explicit reflected maps on a fixed two-sided
collar. The bound is independent of the reciprocals of the target radii. -/
open Set Function Filter Metric Complex
open scoped Topology ComplexConjugate
noncomputable section
namespace CircleDomainRigidity
lemma exists_moving_circle_reflection_limit {U:Set ℂ} {f:ℕ → ℂ → ℂ}
{c:ℕ → ℂ} {r:ℕ → ℝ} {M:ℝ}
(hU:IsOpen U) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U)
(hr:∀n,0<r n)
(hc:∀n,ContinuousOn (f n) (U∩{z | 1≤‖z‖}))
(hd:∀n z,z∈U → 1<‖z‖ → DifferentiableAt ℂ (f n) z)
(ho:∀n z,z∈U → 1≤‖z‖ → r n≤‖f n z-c n‖)
(hb:∀n z,z∈U → ‖z‖=1 → ‖f n z-c n‖=r n)
(hM:∀n,‖c n‖+r n≤M)
(hf:∀n z,z∈U → 1≤‖z‖ → ‖f n z‖≤M) :
∃s:ℕ → ℕ, ∃F:ℂ → ℂ,StrictMono s ∧
TendstoLocallyUniformlyOn (fun n => circleReflectionExtension (c (s n)) (r (s n)) (f (s n)))
F atTop U ∧ DifferentiableOn ℂ F U := by
have hhol (n:ℕ) : DifferentiableOn ℂ (circleReflectionExtension (c n) (r n) (f n)) U :=
circleReflectionExtension_differentiableOn (hr n) hU hz hs (hc n) (hd n) (ho n) (hb n)
obtain ⟨s,F,hsm,hlim⟩ := montelExhaustion (fun _ => U)
(fun n => circleReflectionExtension (c n) (r n) (f n)) (fun _ => hU)
(fun _ _ _ => Subset.refl _) hhol (by
intro K hK hKU
refine ⟨M,?_⟩
intro n z _ hzU
exact circleReflectionExtension_norm_le (hr n) hz hs (hM n)
(fun z hzU hzo => ⟨ho n z hzU hzo,hf n z hzU hzo⟩) hzU)
simp only [iUnion_const] at hlim
exact ⟨s,F,hsm,hlim,hlim.differentiableOn (Eventually.of_forall (fun n => hhol (s n))) hU⟩
lemma moving_circle_reflection_limit_boundary {U:Set ℂ} {f:ℕ → ℂ → ℂ}
{c:ℕ → ℂ} {r:ℕ → ℝ} {s:ℕ → ℕ} {F:ℂ → ℂ} {a:ℂ} {ρ:ℝ}
(hr:∀n,0<r n) (hs:StrictMono s)
(hlim:TendstoLocallyUniformlyOn
(fun n => circleReflectionExtension (c (s n)) (r (s n)) (f (s n))) F atTop U)
(hc:Tendsto c atTop (𝓝 a)) (hρ:Tendsto r atTop (𝓝 ρ))
(hb:∀n z,z∈U → ‖z‖=1 → ‖f n z-c n‖=r n)
{z:ℂ} (hz:z∈U) (hz1:‖z‖=1) : ‖F z-a‖=ρ := by
have hp := hlim.tendsto_at hz
have hnorm := (hp.sub (hc.comp hs.tendsto_atTop)).norm
have he (n:ℕ) :
‖circleReflectionExtension (c (s n)) (r (s n)) (f (s n)) z-c (s n)‖=r (s n) := by
rw [circleReflectionExtension_outer (hr (s n)) hz1.ge]
exact hb (s n) z hz hz1
apply tendsto_nhds_unique hnorm
exact (hρ.comp hs.tendsto_atTop).congr' (Eventually.of_forall (fun n => (he n).symm))
end CircleDomainRigidity
end
end OAI
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