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a97875e | 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 | import OAI.Analysis.CircleDomains.Transfer.RegularC1Chart
namespace OAI
/-! Regular scalar coordinates with both ambient spaces represented by ℂ. -/
open Set Function
open scoped Topology ContDiff
noncomputable section
namespace CircleDomainRigidity
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
end CircleDomainRigidity
end
end OAI
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