openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /ComplexRectangleLimit.lean
| import Mathlib.Analysis.Complex.HasPrimitives | |
| import OAI.Geometry.Cannon.Beltrami.CauchySymbol | |
| namespace OAI | |
| /-! Rectangular Morera passage for smooth maps with vanishing antiholomorphic | |
| error. Uniform convergence is needed only on the four compact edges. -/ | |
| open Set Function Filter MeasureTheory Complex | |
| open scoped Topology Interval | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open CannonBeltrami | |
| def complexRectangleFlux (f : ℂ → ℂ) (z w : ℂ) : ℂ := | |
| (∫x : ℝ in z.re..w.re,f (x+z.im*I))-(∫x : ℝ in z.re..w.re,f (x+w.im*I))+ | |
| I*(∫y : ℝ in z.im..w.im,f (w.re+y*I))-I*(∫y : ℝ in z.im..w.im,f (z.re+y*I)) | |
| lemma locally_uniform_curve_integral {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} | |
| (hf : ∀n,Continuous (f n)) (hl : TendstoLocallyUniformly f g atTop) | |
| (γ : ℝ → ℂ) (hγ : Continuous γ) (a b : ℝ) : | |
| Tendsto (fun n => ∫t in a..b,f n (γ t)) atTop (𝓝 (∫t in a..b,g (γ t))) := by | |
| have ht := tendstoLocallyUniformly_iff_forall_isCompact.mp hl | |
| (γ '' uIcc a b) (isCompact_uIcc.image hγ) | |
| have hc : TendstoUniformlyOn (fun n t => f n (γ t)) (fun t => g (γ t)) atTop (uIcc a b) := | |
| (ht.comp γ).mono (fun t ht => mem_image_of_mem γ ht) | |
| exact hc.tendsto_intervalIntegral_of_continuousOn (.of_forall fun n => (hf n |>.comp hγ).continuousOn) | |
| lemma complexRectangleFlux_tendsto {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} | |
| (hf : ∀n,Continuous (f n)) (hl : TendstoLocallyUniformly f g atTop) (z w : ℂ) : | |
| Tendsto (fun n => complexRectangleFlux (f n) z w) atTop (𝓝 (complexRectangleFlux g z w)) := by | |
| have h₁ := locally_uniform_curve_integral hf hl (fun x => x+z.im*I) (by fun_prop) z.re w.re | |
| have h₂ := locally_uniform_curve_integral hf hl (fun x => x+w.im*I) (by fun_prop) z.re w.re | |
| have h₃ := locally_uniform_curve_integral hf hl (fun y => w.re+y*I) (by fun_prop) z.im w.im | |
| have h₄ := locally_uniform_curve_integral hf hl (fun y => z.re+y*I) (by fun_prop) z.im w.im | |
| exact ((h₁.sub h₂).add (h₃.const_mul I)).sub (h₄.const_mul I) | |
| lemma complexRectangleFlux_eq_bar_integral {f : ℂ → ℂ} | |
| (hf : ContDiff ℝ 1 f) (z w : ℂ) : | |
| complexRectangleFlux f z w=(2*I)*(∫x : ℝ in z.re..w.re, | |
| ∫y : ℝ in z.im..w.im,partialBar f (x+y*I)) := by | |
| have hi : IntegrableOn (fun p => I • fderiv ℝ f p 1-fderiv ℝ f p I) | |
| (uIcc z.re w.re ×ℂ uIcc z.im w.im) := | |
| (((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const).const_smul I |>.sub | |
| ((hf.continuous_fderiv (by norm_num)).clm_apply continuous_const)).continuousOn.integrableOn_compact | |
| (isCompact_uIcc.reProdIm isCompact_uIcc) | |
| have he := integral_boundary_rect_of_differentiableOn_real f z w | |
| (hf.differentiable (by norm_num)).differentiableOn hi | |
| have hid (p : ℂ) : I • fderiv ℝ f p 1-fderiv ℝ f p I=(2*I)*partialBar f p := by | |
| simp only [smul_eq_mul,partialBar] | |
| ring_nf | |
| simp [I_sq] | |
| ring | |
| change complexRectangleFlux f z w=_ at he | |
| rw [he] | |
| simp_rw [hid,intervalIntegral.integral_const_mul] | |
| lemma differentiableOn_of_vanishing_rectangle_errors {U : Set ℂ} (hU : IsOpen U) | |
| {f : ℕ → ℂ → ℂ} {g : ℂ → ℂ} (hf : ∀n,ContDiff ℝ 1 (f n)) | |
| (hl : TendstoLocallyUniformly f g atTop) | |
| (he : ∀z w,Rectangle z w⊆U → | |
| Tendsto (fun n => ∫x : ℝ in z.re..w.re, | |
| ∫y : ℝ in z.im..w.im,partialBar (f n) (x+y*I)) atTop (𝓝 0)) : | |
| DifferentiableOn ℂ g U := by | |
| apply (isConservativeOn_and_continuousOn_iff_isDifferentiableOn hU).mp | |
| refine ⟨?_,(hl.continuous (.of_forall fun n => (hf n).continuous)).continuousOn⟩ | |
| intro z w hzw | |
| have hz : complexRectangleFlux g z w=0 := by | |
| apply tendsto_nhds_unique (complexRectangleFlux_tendsto (fun n => (hf n).continuous) hl z w) | |
| have ht := (he z w hzw).const_mul (2*I) | |
| simpa only [complexRectangleFlux_eq_bar_integral (hf _),mul_zero] using ht | |
| rw [←add_eq_zero_iff_eq_neg,wedgeIntegral_add_wedgeIntegral_eq] | |
| exact hz | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |