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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 | import OAI.Analysis.CircleDomains.Reflection.ModulusBoundaryContinuation
namespace OAI
/-! The lower modulus bound in the logarithm argument is automatic
locally from the actual boundary trace. -/
open Set Function Filter Metric Complex
open scoped Topology
noncomputable section
namespace CircleDomainRigidity
lemma exists_local_holomorphic_continuation_of_modulus_trace {R : ℝ} {f : ℂ → ℂ}
(hR : 0<R) (hd : DifferentiableOn ℂ f (upperSemidisk R))
(hupper : ∀z∈upperSemidisk R,‖f z‖<1)
(htrace : ∀p:ℂ,‖p‖≤R → p.im=0 →
Tendsto (fun z => ‖f z‖) (𝓝[upperSemidisk R] p) (𝓝 1)) :
∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧
DifferentiableOn ℂ F V ∧ EqOn F f (V∩{z : ℂ | 0<z.im}) := by
have ht := htrace 0 (by simpa using hR.le) (by simp)
have he : ∀ᶠ z in 𝓝[upperSemidisk R] (0:ℂ),(1/2:ℝ)<‖f z‖ :=
ht.eventually (eventually_gt_nhds (by norm_num : (1/2:ℝ)<1))
obtain ⟨δ,hδ,hδbound⟩ := Metric.eventually_nhds_iff.mp (eventually_nhdsWithin_iff.mp he)
let r := min R δ/2
have hr : 0<r := by dsimp [r];positivity
have hrR : r≤R := by dsimp [r];nlinarith [min_le_left R δ]
have hrδ : r<δ := by dsimp [r];nlinarith [min_le_right R δ]
have hlower : ∀z∈upperSemidisk r,(1/2:ℝ)≤‖f z‖ := by
intro z hz
apply le_of_lt
apply hδbound
· simpa only [dist_zero_right] using (mem_ball_zero_iff.mp hz.1).trans hrδ
· exact upperSemidisk_mono hrR hz
have htr : ∀p:ℂ,‖p‖≤r → p.im=0 →
Tendsto (fun z => ‖f z‖) (𝓝[upperSemidisk r] p) (𝓝 1) := by
intro p hp hp0
exact (htrace p (hp.trans hrR) hp0).mono_left
(nhdsWithin_mono p (upperSemidisk_mono hrR))
obtain ⟨V,F,hV,h0,hVr,hFd,hFe⟩ := exists_holomorphic_continuation_of_modulus_trace
hr (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)≤1)
(hd.mono (upperSemidisk_mono hrR)) hlower
(fun z hz => hupper z (upperSemidisk_mono hrR hz)) htr
exact ⟨V,F,hV,h0,hVr.trans (ball_subset_ball (by linarith)),hFd,hFe⟩
end CircleDomainRigidity
end
end OAI
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