From 82fdc5ac37fd6986c41bf101715f49cb383f98cd Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Fri, 17 May 2024 10:09:14 +0200 Subject: [PATCH] Update complexHarmonic.lean --- Nevanlinna/complexHarmonic.lean | 192 +++++++++++--------------------- 1 file changed, 62 insertions(+), 130 deletions(-) diff --git a/Nevanlinna/complexHarmonic.lean b/Nevanlinna/complexHarmonic.lean index 57519f2..eca98e5 100644 --- a/Nevanlinna/complexHarmonic.lean +++ b/Nevanlinna/complexHarmonic.lean @@ -26,6 +26,17 @@ def Harmonic (f : ℂ → F) : Prop := (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0) +theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : ℂ → F₁} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) : + Harmonic (f₁ + f₂) := by + constructor + · exact ContDiff.add h₁.1 h₂.1 + · rw [laplace_add h₁.1 h₂.1] + simp + intro z + rw [h₁.2 z, h₂.2 z] + simp + + theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G} (h : Harmonic f) : Harmonic (l ∘ f) := by @@ -138,151 +149,72 @@ theorem log_normSq_of_holomorphic_is_harmonic (h₃ : ∀ z, f z ∈ Complex.slitPlane) : Harmonic (Real.log ∘ Complex.normSq ∘ f) := by + -- Suffices to show Harmonic (⇑Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f) + let F := Real.log ∘ Complex.normSq ∘ f + have : Harmonic (Complex.ofRealCLM ∘ F) → Harmonic F := by + intro hyp + have t₁ : Harmonic (Complex.reCLM ∘ Complex.ofRealCLM ∘ F) := harmonic_comp_CLM_is_harmonic hyp + have t₂ : Complex.reCLM ∘ Complex.ofRealCLM ∘ F = F := rfl + rw [t₂] at t₁ + exact t₁ + apply this + dsimp [F] + -- Suffices to show Harmonic (Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f) + have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by + unfold Function.comp + funext z + apply Complex.ofReal_log + exact Complex.normSq_nonneg (f z) + rw [this] - /- We start with a number of lemmas on regularity of all the functions involved -/ - - -- The norm square is real C² - have normSq_is_real_C2 : ContDiff ℝ 2 Complex.normSq := by - unfold Complex.normSq + -- Suffices to show Harmonic (Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f)) + have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ℂ) ∘ f) * f := by + funext z simp - conv => - arg 3 - intro x - rw [← Complex.reCLM_apply, ← Complex.imCLM_apply] - apply ContDiff.add - apply ContDiff.mul - apply ContinuousLinearMap.contDiff Complex.reCLM - apply ContinuousLinearMap.contDiff Complex.reCLM - apply ContDiff.mul - apply ContinuousLinearMap.contDiff Complex.imCLM - apply ContinuousLinearMap.contDiff Complex.imCLM + exact Complex.normSq_eq_conj_mul_self + rw [this] - -- f is real C² - have f_is_real_C2 : ContDiff ℝ 2 f := - ContDiff.restrict_scalars ℝ (Differentiable.contDiff h₁) - -- Complex.log ∘ f is real C² - have log_f_is_holomorphic : Differentiable ℂ (Complex.log ∘ f) := by - intro z - apply DifferentiableAt.comp - exact Complex.differentiableAt_log (h₃ z) - exact h₁ z - - -- Real.log |f|² is real C² - have t₄ : ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f) := by - rw [contDiff_iff_contDiffAt] - intro z - apply ContDiffAt.comp - apply Real.contDiffAt_log.mpr + -- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f) + -- THIS IS WHERE WE USE h₃ + have : Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f := by + unfold Function.comp + funext z simp + rw [Complex.log_mul_eq_add_log_iff] + + have : Complex.arg ((starRingEnd ℂ) (f z)) = - Complex.arg (f z) := by + rw [Complex.arg_conj] + have : ¬ Complex.arg (f z) = Real.pi := by + exact Complex.slitPlane_arg_ne_pi (h₃ z) + simp + tauto + rw [this] + simp + constructor + · exact Real.pi_pos + · exact Real.pi_nonneg + exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z) exact h₂ z - apply ContDiff.comp_contDiffAt z normSq_is_real_C2 - exact ContDiff.contDiffAt f_is_real_C2 + rw [this] - have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by + apply harmonic_add_harmonic_is_harmonic + have : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by funext z unfold Function.comp rw [Complex.log_conj] rfl exact Complex.slitPlane_arg_ne_pi (h₃ z) + rw [this] + rw [← harmonic_iff_comp_CLE_is_harmonic] - constructor - · -- logabs f is real C² - have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by - funext z - simp - unfold Complex.abs - simp - rw [Real.log_sqrt] - rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2] - exact Complex.normSq_nonneg (f z) - rw [this] - - have : (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : ℝ)⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by - exact rfl - rw [this] - apply ContDiff.const_smul - exact t₄ - - · -- Laplace vanishes - have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by - funext z - simp - unfold Complex.abs - simp - rw [Real.log_sqrt] - rw [div_eq_inv_mul (Real.log (Complex.normSq (f z))) 2] - exact Complex.normSq_nonneg (f z) - rw [this] - rw [laplace_smul] - simp - - have : ∀ (z : ℂ), Complex.laplace (Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 ↔ Complex.laplace (Complex.ofRealCLM ∘ Real.log ∘ ⇑Complex.normSq ∘ f) z = 0 := by - intro z - rw [laplace_compContLin] - simp - -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f) - exact t₄ - conv => - intro z - rw [this z] - - have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ f = Complex.log ∘ Complex.ofRealCLM ∘ Complex.normSq ∘ f := by - unfold Function.comp - funext z - apply Complex.ofReal_log - exact Complex.normSq_nonneg (f z) - rw [this] - - have : Complex.ofRealCLM ∘ ⇑Complex.normSq ∘ f = ((starRingEnd ℂ) ∘ f) * f := by - funext z - simp - exact Complex.normSq_eq_conj_mul_self - rw [this] - - have : Complex.log ∘ (⇑(starRingEnd ℂ) ∘ f * f) = Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f + Complex.log ∘ f := by - unfold Function.comp - funext z - simp - rw [Complex.log_mul_eq_add_log_iff] - - have : Complex.arg ((starRingEnd ℂ) (f z)) = - Complex.arg (f z) := by - rw [Complex.arg_conj] - have : ¬ Complex.arg (f z) = Real.pi := by - exact Complex.slitPlane_arg_ne_pi (h₃ z) - simp - tauto - rw [this] - simp - constructor - · exact Real.pi_pos - · exact Real.pi_nonneg - exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z) - exact h₂ z - rw [this] - rw [laplace_add] - - rw [t₂, laplace_compCLE] + repeat + apply holomorphic_is_harmonic intro z - simp - rw [(holomorphic_is_harmonic log_f_is_holomorphic).2 z] - simp - - -- ContDiff ℝ 2 (Complex.log ∘ f) - exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic) - - -- ContDiff ℝ 2 (Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f) - rw [t₂] - apply ContDiff.comp - exact ContinuousLinearEquiv.contDiff Complex.conjCLE - exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic) - - -- ContDiff ℝ 2 (Complex.log ∘ f) - exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff log_f_is_holomorphic) - - -- ContDiff ℝ 2 (Real.log ∘ ⇑Complex.normSq ∘ f) - exact t₄ + apply DifferentiableAt.comp + exact Complex.differentiableAt_log (h₃ z) + exact h₁ z theorem logabs_of_holomorphic_is_harmonic