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ddc84c5 | 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 72 73 | import Mathlib
import OAI.Analysis.Crouzeix.Univalent
namespace OAI
/-! Boundary Modulus. -/
noncomputable section
open Set Filter Metric Topology Function
namespace CrouzeixHilbert.Conformal
attribute [local instance] Classical.propDecidable
theorem isCompact_preimage_closedBall {U : Set ℂ} (hU : IsOpen U)
{f : ℂ → ℂ} (hf : DifferentiableOn ℂ f U) (hbij : BijOn f U (ball 0 1))
{r : ℝ} (hr : r < 1) : IsCompact (U ∩ f ⁻¹' closedBall 0 r) := by
have hsub : closedBall (0 : ℂ) r ⊆ ball 0 1 := closedBall_subset_ball hr
have hd : DifferentiableOn ℂ (invFunOn f U) (ball 0 1) :=
fun w hw => (hasStrictDerivAt_invFunOn hU hf hbij hw).hasDerivAt.differentiableAt.differentiableWithinAt
have he : U ∩ f ⁻¹' closedBall 0 r = invFunOn f U '' closedBall 0 r := by
ext z
constructor
· rintro ⟨hzu, hz⟩
exact ⟨f z, hz, hbij.invOn_invFunOn.1 hzu⟩
· rintro ⟨w, hw, rfl⟩
exact ⟨(hbij.symm hbij.invOn_invFunOn.symm).mapsTo (hsub hw),
by simpa only [mem_preimage, hbij.invOn_invFunOn.2 (hsub hw)] using hw⟩
rw [he]
exact (isCompact_closedBall 0 r).image_of_continuousOn (hd.continuousOn.mono hsub)
theorem tendsto_norm_one_at_frontier {U : Set ℂ} (hU : IsOpen U)
{f : ℂ → ℂ} (hf : DifferentiableOn ℂ f U) (hbij : BijOn f U (ball 0 1))
{p : ℂ} (hp : p ∈ frontier U) :
Tendsto (fun z => ‖f z‖) (𝓝[U] p) (𝓝 1) := by
apply Metric.tendsto_nhds.mpr
intro ε hε
have hK := isCompact_preimage_closedBall hU hf hbij (sub_lt_self (1 : ℝ) hε)
have hpn : p ∉ U ∩ f ⁻¹' closedBall 0 (1 - ε) := by
intro he
exact hp.2 (hU.interior_eq.symm ▸ he.1)
filter_upwards [Filter.mem_inf_of_left (hK.isClosed.isOpen_compl.mem_nhds hpn),
self_mem_nhdsWithin] with z hz₁ hz₂
have hsmall : ‖f z‖ < 1 := mem_ball_zero_iff.mp (hbij.mapsTo hz₂)
have hlarge : 1 - ε < ‖f z‖ := by
by_contra h
exact hz₁ ⟨hz₂, mem_closedBall_zero_iff.mpr (le_of_not_gt h)⟩
rw [Real.dist_eq, abs_of_nonpos (by linarith)]
linarith
theorem continuousOn_closed_modulus {U : Set ℂ} (hU : IsOpen U)
{f : ℂ → ℂ} (hf : DifferentiableOn ℂ f U) (hbij : BijOn f U (ball 0 1)) :
ContinuousOn (fun z => if z ∈ U then ‖f z‖ else 1) (closure U) := by
classical
intro p hp
by_cases hpU : p ∈ U
· apply (hf.continuousOn.norm.continuousAt (hU.mem_nhds hpU)).continuousWithinAt.congr_of_eventuallyEq
· filter_upwards [Filter.mem_inf_of_left (hU.mem_nhds hpU)] with z hz
simp only [ite_eq_left hz]
· simp only [ite_eq_left hpU]
· have hpf : p ∈ frontier U := ⟨hp, by simpa only [hU.interior_eq] using hpU⟩
have ht := tendsto_norm_one_at_frontier hU hf hbij hpf
rw [ContinuousWithinAt, ite_eq_right hpU]
let : DecidablePred U := fun z => Classical.propDecidable _
exact (ht.mono_left (nhdsWithin_mono p inter_subset_right)).if_nhdsWithin tendsto_const_nhds
end CrouzeixHilbert.Conformal
end
end OAI
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