openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /ConformalLocalInverse.lean
| import OAI.Analysis.CircleDomains.Sobolev.SmoothSphereCharts | |
| namespace OAI | |
| /-! | |
| # Local inverses in sphere charts | |
| On an open set the chart expressions of a conformal sphere map are | |
| analytic. The ordinary complex inverse function theorem therefore | |
| identifies its neighborhood filters. An injective conformal map then | |
| has a conformal inverse on its image, with no global extension assumed. | |
| -/ | |
| open Set Filter Function | |
| open scoped Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| theorem IsConformalOn.analyticAt_sphereLocalMap | |
| {f : Sphere → Sphere} {U : Set Sphere} (hf : IsConformalOn f U) | |
| (hU : IsOpen U) {p : Sphere} (hp : p ∈ U) : | |
| AnalyticAt ℂ (sphereLocalMap f p) (chart p p) := by | |
| let g : Sphere → Sphere := chartTransform (f p) ∘ f ∘ chartTransform p | |
| let V : Set Sphere := (chartTransform p) ⁻¹' U | |
| have hV : IsOpen V := hU.preimage (chartTransform p).continuous | |
| have hg : IsConformalOn g V := by | |
| intro q hq | |
| exact (isConformalAt_chartTransform _ _).comp | |
| ((hf _ hq).comp (isConformalAt_chartTransform _ q)) | |
| have hcenter : chartTransform p ((chart p p : ℂ) : Sphere) = p := by | |
| rw [← chartSymm_eq_transform] | |
| exact chartSymm_chart_self p | |
| have hzV : ((chart p p : ℂ) : Sphere) ∈ V := by | |
| change chartTransform p ((chart p p : ℂ) : Sphere) ∈ U | |
| rwa [hcenter] | |
| have hfin : g ((chart p p : ℂ) : Sphere) ≠ OnePoint.infty := by | |
| change chartTransform (f p) (f (chartTransform p ((chart p p : ℂ) : Sphere))) ≠ _ | |
| rw [hcenter] | |
| exact chartTransform_center_ne_infty _ | |
| have ha := analyticAt_finite_map_of_conformal hV hg hzV hfin | |
| have heq : (fun w : ℂ => finiteCoord (g (w : Sphere))) = sphereLocalMap f p := by | |
| funext w | |
| rw [sphereLocalMap, chart_eq_coordinate_transform, chartSymm_eq_transform] | |
| rfl | |
| rwa [heq] at ha | |
| theorem map_chartSymm_nhds (p : Sphere) : | |
| map (chartSymm p) (𝓝 (chart p p)) = 𝓝 p := by | |
| have he : chartSymm p = fun z : ℂ => chartTransform p (z : Sphere) := | |
| funext (chartSymm_eq_transform p) | |
| rw [he] | |
| change map (chartTransform p ∘ ((↑) : ℂ → Sphere)) (𝓝 (chart p p)) = _ | |
| rw [← map_map, ← OnePoint.nhds_coe_eq, (chartTransform p).map_nhds_eq] | |
| congr 1 | |
| rw [← chartSymm_eq_transform, chartSymm_chart_self] | |
| theorem IsConformalOn.map_nhds_eq | |
| {f : Sphere → Sphere} {U : Set Sphere} (hf : IsConformalOn f U) | |
| (hU : IsOpen U) {p : Sphere} (hp : p ∈ U) : map f (𝓝 p) = 𝓝 (f p) := by | |
| obtain ⟨hc, d, hd, hder⟩ := hf p hp | |
| have ha : ContDiffAt ℂ 1 (sphereLocalMap f p) (chart p p) := | |
| (hf.analyticAt_sphereLocalMap hU hp).contDiffAt | |
| have hstrict : HasStrictDerivAt (sphereLocalMap f p) d (chart p p) := | |
| ha.hasStrictDerivAt' hder (by norm_num) | |
| have hcenter : sphereLocalMap f p (chart p p) = chart (f p) (f p) := by | |
| simp only [sphereLocalMap, chartSymm_chart_self] | |
| have heq : (f ∘ chartSymm p) =ᶠ[𝓝 (chart p p)] | |
| (chartSymm (f p) ∘ sphereLocalMap f p) := by | |
| have ht : Tendsto (f ∘ chartSymm p) (𝓝 (chart p p)) (𝓝 (f p)) := by | |
| have hc' : ContinuousAt f (chartSymm p (chart p p)) := by | |
| rwa [chartSymm_chart_self] | |
| simpa only [Function.comp_apply, chartSymm_chart_self] using | |
| (hc'.comp (continuous_chartSymm p).continuousAt).tendsto | |
| filter_upwards [ht.eventually (eventually_chartSymm_chart (f p))] with z hz | |
| exact hz.symm | |
| calc | |
| map f (𝓝 p) = map (f ∘ chartSymm p) (𝓝 (chart p p)) := by | |
| rw [← map_map, map_chartSymm_nhds] | |
| _ = map (chartSymm (f p) ∘ sphereLocalMap f p) (𝓝 (chart p p)) := map_congr heq | |
| _ = map (chartSymm (f p)) (𝓝 (chart (f p) (f p))) := by | |
| rw [← map_map, hstrict.map_nhds_eq hd, hcenter] | |
| _ = 𝓝 (f p) := map_chartSymm_nhds (f p) | |
| theorem IsConformalOn.isOpen_image | |
| {f : Sphere → Sphere} {U O : Set Sphere} (hf : IsConformalOn f U) | |
| (hU : IsOpen U) (hO : IsOpen O) (hOU : O ⊆ U) : IsOpen (f '' O) := by | |
| apply isOpen_iff_mem_nhds.mpr | |
| rintro _ ⟨p, hp, rfl⟩ | |
| rw [← hf.map_nhds_eq hU (hOU hp)] | |
| exact image_mem_map (hO.mem_nhds hp) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |