| import OAI.Analysis.CircleDomains.Transfer.RegularC1Chart |
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| namespace OAI |
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| /-! Regular scalar coordinates with both ambient spaces represented by ℂ. -/ |
| open Set Function |
| open scoped Topology ContDiff |
| noncomputable section |
| namespace CircleDomainRigidity |
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| lemma exists_complex_c1_level_chart {D : Set ℂ} (hD : IsOpen D) |
| {f : ℂ → ℝ} (hf : ContDiffOn ℝ 1 f D) {z : ℂ} (hzD : z ∈ D) |
| (hz : fderiv ℝ f z ≠ 0) : |
| ∃ e : OpenPartialHomeomorph ℂ ℂ, z ∈ e.source ∧ e.source ⊆ D ∧ |
| (∀ w, (e w).re = f w) ∧ ContDiffOn ℝ 1 e e.source ∧ |
| ContDiffOn ℝ 1 e.symm e.target := by |
| obtain ⟨e, hz, heD, he, heC, hiC⟩ := exists_c1_planar_level_chart hD hf hzD hz |
| let g := e.transHomeomorph Complex.equivRealProdCLM.symm.toHomeomorph |
| refine ⟨g, hz, heD, ?_, ?_, ?_⟩ |
| · intro w |
| exact he w |
| · change ContDiffOn ℝ 1 (Complex.equivRealProdCLM.symm ∘ e) e.source |
| exact Complex.equivRealProdCLM.symm.contDiff.comp_contDiffOn heC |
| · change ContDiffOn ℝ 1 (e.symm ∘ Complex.equivRealProdCLM) g.target |
| apply hiC.comp Complex.equivRealProdCLM.contDiff.contDiffOn |
| intro y hy |
| exact hy |
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|
| end CircleDomainRigidity |
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| end |
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| end OAI |
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