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