| 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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|
