| import OAI.Analysis.CircleDomains.Topology.LevelChartJordanSides |
|
|
| namespace OAI |
|
|
| /-! Symmetric coordinate boxes can be chosen inside any prescribed |
| ambient neighborhood of a regular level point. -/ |
| open Set Function Filter Metric |
| open scoped Topology |
| noncomputable section |
| namespace CircleDomainRigidity |
| |
| lemma exists_openPartialHomeomorph_box |
| (e : OpenPartialHomeomorph ℂ (ℝ × ℝ)) (hx : x ∈ e.source) |
| (hU : IsOpen U) (hxU : x ∈ U) : |
| ∃ r > 0, Ioo ((e x).1-r) ((e x).1+r) ×ˢ Ioo ((e x).2-r) ((e x).2+r) ⊆ |
| e.target ∩ e.symm ⁻¹' U := by |
| have hW := e.isOpen_inter_preimage_symm hU |
| have hxW : e x ∈ e.target ∩ e.symm ⁻¹' U := |
| ⟨e.map_source hx, by simpa only [mem_preimage, e.left_inv hx] using hxU⟩ |
| obtain ⟨r, hr, hsub⟩ := Metric.isOpen_iff.mp hW _ hxW |
| refine ⟨r, hr, ?_⟩ |
| intro q hq |
| apply hsub |
| rw [mem_ball, Prod.dist_eq, max_lt_iff, Real.dist_eq, Real.dist_eq] |
| constructor <;> rw [abs_lt] <;> constructor <;> linarith [hq.1.1, hq.1.2, hq.2.1, hq.2.2] |
|
|
| lemma regular_jordan_local_side |
| (H : ℂ ≃ₜ ℂ) |
| (hf : ContDiff ℝ (⊤ : ℕ∞) f) (ht : IsPlanarRegularValue f t) |
| (hlevel : frontier (jordanInterior H) ⊆ ) |
| (hx : x ∈ frontier (jordanInterior H)) : |
| ∃ N : Set ℂ, IsOpen N ∧ x ∈ N ∧ |
| ((∀ y ∈ N, t < f y → y ∈ jordanInterior H) ∨ |
| (∀ y ∈ N, f y < t → y ∈ jordanInterior H)) := by |
| obtain ⟨e, hxe, he, _, _⟩ := exists_smooth_planar_level_chart hf (ht x (hlevel hx)) |
| obtain ⟨r, hr, hbox⟩ := exists_openPartialHomeomorph_box e hxe isOpen_univ (mem_univ x) |
| have hx1 : (e x).1 = t := (he x).trans (hlevel hx) |
| rw [hx1] at hbox |
| let B := Ioo (t-r) (t+r) ×ˢ Ioo ((e x).2-r) ((e x).2+r) |
| let N := e.symm '' B |
| have hBt : B ⊆ e.target := fun _ hy => (hbox hy).1 |
| refine ⟨N, e.symm.isOpen_image_of_subset_source (isOpen_Ioo.prod isOpen_Ioo) hBt, |
| ⟨e x, ⟨⟨by rw [hx1]; linarith, by rw [hx1]; linarith⟩, |
| ⟨by linarith, by linarith⟩⟩, e.left_inv hxe⟩, ?_⟩ |
| rcases level_chart_jordan_sides H f t e he hlevel hx hxe hr hBt with hp | hm |
| · left |
| rintro y ⟨q, hq, rfl⟩ hy |
| apply hp.1 |
| refine ⟨q, ⟨⟨?_, hq.1.2⟩, hq.2⟩, rfl⟩ |
| have heq := congrArg Prod.fst (e.right_inv (hBt hq)) |
| rw [he] at heq |
| linarith |
| · right |
| rintro y ⟨q, hq, rfl⟩ hy |
| apply hm.1 |
| refine ⟨q, ⟨⟨hq.1.1, ?_⟩, hq.2⟩, rfl⟩ |
| have heq := congrArg Prod.fst (e.right_inv (hBt hq)) |
| rw [he] at heq |
| linarith |
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|
| end CircleDomainRigidity |
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| end |
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| end OAI |
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