| import OAI.Analysis.IntegralMeans.Degree |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open MeasureTheory Function |
| open Set Filter |
| open Set MeasureTheory Filter Function |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open TopologicalSpace |
| open scoped CompactlySupported |
| open scoped ENNReal |
| open scoped Manifold |
| open scoped Topology CompactlySupported ComplexConjugate |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open scoped Topology ENNReal NNReal |
| namespace Brennan |
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| lemma exists_smooth_cutoff {K U : Set ℂ} (hK : IsCompact K) (hU : IsOpen U) |
| (hKU : K ⊆ U) : |
| ∃ χ : ℂ → ℝ, ContDiff ℝ 2 χ ∧ HasCompactSupport χ ∧ tsupport χ ⊆ U ∧ |
| (∀ z ∈ K, χ =ᶠ[𝓝 z] fun _ => 1) ∧ ∀ z, χ z ∈ Icc 0 1 := by |
| obtain ⟨L,hL,hLc,hKL,hLU⟩ := exists_compact_closed_between hK hU hKU |
| obtain ⟨χ,hχ1,hχ0,hχrange⟩ := |
| exists_contMDiffMap_one_nhds_of_subset_interior 𝓘(ℝ,ℂ) hK.isClosed hKL (n := 2) |
| have hs : tsupport (χ : ℂ → ℝ) ⊆ L := by |
| apply closure_minimal _ hLc |
| intro z hz |
| by_contra hn |
| exact hz (hχ0 z hn) |
| refine ⟨χ,contMDiff_iff_contDiff.mp χ.contMDiff, |
| hL.of_isClosed_subset (isClosed_tsupport _) hs,hs.trans hLU,?_,hχrange⟩ |
| intro z hz |
| exact hχ1.filter_mono (nhds_le_nhdsSet hz) |
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| lemma det_fderiv_complex {f : ℂ → ℂ} {z : ℂ} (hd : DifferentiableAt ℂ f z) : |
| LinearMap.det (fderiv ℝ f z).toLinearMap = ‖deriv f z‖^2 := by |
| rw [(hd.hasDerivAt.hasFDerivAt.restrictScalars ℝ).fderiv,determinant_complex_real, |
| ← Complex.normSq_eq_norm_sq] |
| simp only [ContinuousLinearMap.coe_coe, ContinuousLinearMap.coe_restrictScalars', |
| ContinuousLinearMap.toSpanSingleton_apply, smul_eq_mul, one_mul, |
| Complex.I_mul, Complex.normSq_apply] |
| ring |
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| lemma smooth_extension_near_compact {K U : Set ℂ} (hK : IsCompact K) |
| (hU : IsOpen U) (hKU : K ⊆ U) {x : ℂ → ℂ} |
| (hx : ∀ z ∈ U, ContDiffAt ℝ 2 x z) : |
| ∃ X : ℂ → ℂ, ContDiff ℝ 2 X ∧ ∀ z ∈ K, X =ᶠ[𝓝 z] x := by |
| obtain ⟨χ,hχc,_hχK,hχU,hχ1,_hχrange⟩ := exists_smooth_cutoff hK hU hKU |
| let X := fun z => (χ z : ℂ)*x z |
| have hX : ContDiff ℝ 2 X := by |
| rw [contDiff_iff_contDiffAt] |
| intro z |
| by_cases hz : z ∈ tsupport χ |
| · exact (Complex.ofRealCLM.contDiff.contDiffAt.comp z hχc.contDiffAt).mul (hx z (hχU hz)) |
| · have he : X =ᶠ[𝓝 z] fun _ => 0 := by |
| filter_upwards [isClosed_tsupport χ |>.isOpen_compl.mem_nhds hz] with w hw |
| have hzero : χ w = 0 := notMem_support.mp (fun h => hw (subset_tsupport χ h)) |
| simp only [X,hzero,Complex.ofReal_zero,zero_mul] |
| exact contDiffAt_const.congr_of_eventuallyEq he |
| refine ⟨X,hX,?_⟩ |
| intro z hz |
| filter_upwards [hχ1 z hz] with w hw |
| simp only [X,hw,Complex.ofReal_one,one_mul] |
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| lemma meromorphic_boundary_winding_le_zero {h H : ℂ → ℂ} |
| (hh : DifferentiableOn ℂ h (Metric.ball 0 1)) |
| (hH : DifferentiableOn ℂ H (Metric.ball 0 1 \ {0})) |
| (hdef : ∀ z ∈ Metric.ball (0 : ℂ) 1, z ≠ 0 → H z = z⁻¹+h z) |
| (hHd : ∀ z ∈ Metric.ball (0 : ℂ) 1 \ {0}, deriv H z ≠ 0) |
| (hHi : InjOn H (Metric.ball 0 1 \ {0})) |
| {r : ℝ} (hr : 0 < r) (hr1 : r < 1) (ξ : ℂ) |
| (hξ : ξ ∉ H '' Metric.sphere 0 r) : |
| windingNumber (⟨fun t => H ((r : ℂ)*circleLoop t)-ξ, by |
| refine Continuous.sub ?_ continuous_const |
| apply hH.continuousOn.comp_continuous (by fun_prop) |
| intro t |
| have he : ‖(r : ℂ)*circleLoop t‖ = r := by simp [abs_of_pos hr] |
| constructor |
| · rw [Metric.mem_ball,dist_zero_right,he] |
| exact hr1 |
| · change (r : ℂ)*circleLoop t ≠ 0 |
| exact norm_pos_iff.mp (by rw [he]; exact hr)⟩) ≤ 0 := by |
| classical |
| let U : Set ℂ := Metric.ball 0 1 |
| let K : Set ℂ := Metric.closedBall 0 r |
| have hKU : K ⊆ U := Metric.closedBall_subset_ball hr1 |
| let x : ℂ → ℂ := fun z => 1+z*(h z-ξ) |
| have hx (z) (hz : z ∈ U) : AnalyticAt ℂ x z := by |
| exact analyticAt_const.add (analyticAt_id.mul ((hh.analyticOnNhd Metric.isOpen_ball z hz).sub analyticAt_const)) |
| obtain ⟨X,hXc,hX⟩ := smooth_extension_near_compact (isCompact_closedBall 0 r) |
| Metric.isOpen_ball hKU (fun z hz => (hx z hz).contDiffAt.restrict_scalars ℝ) |
| have hXval (z) (hz : z ∈ K) : X z = x z := (hX z hz).self_of_nhds |
| have hxzero {z} (hz : z ∈ U) : x z = 0 ↔ z ≠ 0 ∧ H z = ξ := by |
| by_cases hz0 : z = 0 |
| · simp [x,hz0] |
| · rw [hdef z hz hz0] |
| dsimp [x] |
| field_simp |
| constructor <;> intro he |
| · exact ⟨hz0,by linear_combination he⟩ |
| · linear_combination he.2 |
| have hss : {z | z ∈ K ∧ X z = 0}.Subsingleton := by |
| intro z hz w hw |
| obtain ⟨hz0,hzξ⟩ := (hxzero (hKU hz.1)).mp ((hXval z hz.1) ▸ hz.2) |
| obtain ⟨hw0,hwξ⟩ := (hxzero (hKU hw.1)).mp ((hXval w hw.1) ▸ hw.2) |
| exact hHi ⟨hKU hz.1,hz0⟩ ⟨hKU hw.1,hw0⟩ (hzξ.trans hwξ.symm) |
| let s := hss.finite.toFinset |
| have hs (z) : z ∈ s ↔ z ∈ K ∧ X z = 0 := by simp [s] |
| have hder (z) (hz : z ∈ s) : DifferentiableAt ℂ X z ∧ deriv X z ≠ 0 := by |
| have hzK := ((hs z).mp hz).1 |
| have hzero := ((hs z).mp hz).2 |
| obtain ⟨hz0,hzξ⟩ := (hxzero (hKU hzK)).mp (by rwa [hXval z hzK] at hzero) |
| have hUz : z ∈ Metric.ball (0 : ℂ) 1 \ {0} := ⟨hKU hzK,hz0⟩ |
| have hHd' := (hH z hUz).differentiableAt ((Metric.isOpen_ball.sdiff isClosed_singleton).mem_nhds hUz) |
| have heq : x =ᶠ[𝓝 z] fun w => w*(H w-ξ) := by |
| filter_upwards [(Metric.isOpen_ball.sdiff isClosed_singleton).mem_nhds hUz] with w hw |
| have hw0 : w ≠ 0 := hw.2 |
| rw [hdef w hw.1 hw0] |
| dsimp [x] |
| field_simp [hw0] |
| ring |
| have hD : HasDerivAt X (z*deriv H z) z := by |
| have hD := (hasDerivAt_id z).mul (hHd'.hasDerivAt.sub_const ξ) |
| simp only [hzξ,sub_self,mul_zero,zero_add] at hD |
| exact hD.congr_of_eventuallyEq ((hX z hzK).trans heq) |
| exact ⟨hD.differentiableAt,hD.deriv ▸ mul_ne_zero hz0 (hHd z hUz)⟩ |
| let γ : C(unitInterval,ℂ) := (r : ℂ) • circleLoop |
| have hγnorm (t) : ‖γ t‖ = r := by simp [γ,abs_of_pos hr] |
| have hγK (t) : γ t ∈ K := by simpa [K] using (hγnorm t).le |
| have hγU (t) : γ t ∈ U := hKU (hγK t) |
| have hγ0 (t) : γ t ≠ 0 := norm_pos_iff.mp (by rw [hγnorm]; exact hr) |
| have hγξ (t) : H (γ t) ≠ ξ := fun he => hξ ⟨γ t,by simpa using hγnorm t,he⟩ |
| let Y : C(ℂ,ℂ) := ⟨X,hXc.continuous⟩ |
| have hXY (t) : Y (γ t) = γ t*(H (γ t)-ξ) := by |
| rw [show Y (γ t) = X (γ t) from rfl,hXval _ (hγK t),hdef _ (hγU t) (hγ0 t)] |
| dsimp only [x] |
| field_simp [hγ0 t] |
| ring |
| have hγnonzero (t) : Y (γ t) ≠ 0 := by rw [hXY]; exact mul_ne_zero (hγ0 t) (sub_ne_zero.mpr (hγξ t)) |
| have hi := windingNumber_finite_regular_zeros (convex_closedBall (0 : ℂ) r) s Y |
| (fun z => fderiv ℝ X z) (fun z hz => ((hs z).mp hz).1) |
| (fun z hz => by simp [hs,show z ∈ K from hz,Y]) |
| (fun z hz => ⟨((hder z hz).1.restrictScalars ℝ).hasFDerivAt,by |
| rw [det_fderiv_complex (hder z hz).1] |
| exact pow_ne_zero _ (norm_ne_zero_iff.mpr (hder z hz).2)⟩) |
| γ (by simp [γ]) hγK hγnonzero |
| have hins (z) (hz : z ∈ s) : ‖z‖ < r := by |
| have hzK := ((hs z).mp hz).1 |
| have hle : ‖z‖ ≤ r := by simpa [K] using hzK |
| refine lt_of_le_of_ne hle ?_ |
| intro he |
| have hzero := ((hs z).mp hz).2 |
| have hzξ := ((hxzero (hKU hzK)).mp (by rwa [hXval z hzK] at hzero)).2 |
| exact hξ ⟨z,by simpa using he,hzξ⟩ |
| have hi' : windingNumber (Y.comp γ) = (s.card : ℤ) := by |
| rw [hi] |
| calc |
| _ = ∑ _z ∈ s, (1 : ℤ) := by |
| apply Finset.sum_congr rfl |
| intro z hz |
| rw [loopIndex_circle_inside hr (hins z hz),mul_one] |
| simp only [planeIndex,det_fderiv_complex (hder z hz).1, |
| sq_pos_of_pos (norm_pos_iff.mpr (hder z hz).2),ite_true] |
| _ = ↑s.card := by simp |
| have hcard : s.card ≤ 1 := Finset.card_le_one.mpr (by |
| intro z hz w hw |
| exact hss ((hs z).mp hz) ((hs w).mp hw)) |
| let η : C(unitInterval,ℂ) := ⟨fun t => H (γ t)-ξ,by |
| refine Continuous.sub ?_ continuous_const |
| apply hH.continuousOn.comp_continuous γ.continuous |
| exact fun t => ⟨hγU t,hγ0 t⟩⟩ |
| have he : Y.comp γ = γ*η := by ext t; exact hXY t |
| have hwγ : windingNumber γ = 1 := by |
| apply windingNumber_eq_of_logIncrement (by simp [γ]) |
| simpa [γ] using circleLoop_logIncrement (r : ℂ) (by exact_mod_cast hr.ne') |
| rw [he,windingNumber_mul γ η (by simp [γ]) (by simp [η,γ]) hγ0 |
| (fun t => sub_ne_zero.mpr (hγξ t)),hwγ] at hi' |
| change windingNumber η ≤ 0 |
| have hc : (s.card : ℤ) ≤ 1 := by exact_mod_cast hcard |
| omega |
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| lemma det_holomorphic_plus_conjugate {h : ℂ → ℂ} {z : ℂ} |
| (hh : DifferentiableAt ℂ h z) (a : ℝ) : |
| LinearMap.det (fderiv ℝ (fun w => a • conj w+h w) z).toLinearMap = |
| ‖deriv h z‖^2-a^2 := by |
| have hD := (Complex.conjCLE.hasFDerivAt.const_smul a).add |
| (hh.hasDerivAt.hasFDerivAt.restrictScalars ℝ) |
| change HasFDerivAt (fun w => a • conj w+h w) _ z at hD |
| rw [hD.fderiv,determinant_complex_real,← Complex.normSq_eq_norm_sq] |
| simp only [ContinuousLinearMap.coe_coe,add_apply, |
| smul_apply,ContinuousLinearMap.coe_restrictScalars', |
| ContinuousLinearMap.toSpanSingleton_apply, |
| smul_eq_mul,one_mul,Complex.real_smul,Complex.normSq_apply] |
| simp [Complex.add_re,Complex.add_im,Complex.mul_re,Complex.mul_im] |
| ring |
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| lemma meromorphic_univalent_area {h H : ℂ → ℂ} |
| (hh : DifferentiableOn ℂ h (Metric.ball 0 1)) |
| (hH : DifferentiableOn ℂ H (Metric.ball 0 1 \ {0})) |
| (hdef : ∀ z ∈ Metric.ball (0 : ℂ) 1, z ≠ 0 → H z = z⁻¹+h z) |
| (hHd : ∀ z ∈ Metric.ball (0 : ℂ) 1 \ {0}, deriv H z ≠ 0) |
| (hHi : InjOn H (Metric.ball 0 1 \ {0})) |
| {r : ℝ} (hr : 0 < r) (hr1 : r < 1) : |
| ∫ z in Metric.closedBall 0 r, ‖deriv h z‖^2 ∂volume ≤ |
| volume.real (Metric.closedBall (0 : ℂ) r) * (r^2)⁻¹^2 := by |
| let K : Set ℂ := Metric.closedBall 0 r |
| let x : ℂ → ℂ := fun z => (r^2)⁻¹ • conj z+h z |
| have hKU : K ⊆ Metric.ball 0 1 := Metric.closedBall_subset_ball hr1 |
| have hx (z) (hz : z ∈ Metric.ball (0 : ℂ) 1) : ContDiffAt ℝ 2 x z := by |
| exact (Complex.conjCLE.contDiff.contDiffAt.const_smul (r^2)⁻¹).add |
| ((hh.analyticOnNhd Metric.isOpen_ball z hz).contDiffAt.restrict_scalars ℝ) |
| obtain ⟨X,hXc,hX⟩ := smooth_extension_near_compact (isCompact_closedBall 0 r) |
| Metric.isOpen_ball hKU hx |
| have hXval (z) (hz : z ∈ K) : X z = x z := (hX z hz).self_of_nhds |
| have hb (z) (hz : z ∈ Metric.sphere (0 : ℂ) r) : X z = H z := by |
| have hn : ‖z‖ = r := by simpa using hz |
| have hz0 : z ≠ 0 := norm_pos_iff.mp (by rw [hn];exact hr) |
| rw [hXval z (Metric.sphere_subset_closedBall hz),hdef z (hKU (Metric.sphere_subset_closedBall hz)) hz0] |
| dsimp [x] |
| rw [Complex.inv_def,Complex.normSq_eq_norm_sq,hn] |
| congr 1 |
| simp [mul_comm] |
| have hw (ξ) (hξ : ξ ∉ X '' Metric.sphere 0 r) : |
| windingNumber (⟨fun t => X ((r : ℂ)*circleLoop t)-ξ,by fun_prop⟩) ≤ 0 := by |
| have hξ' : ξ ∉ H '' Metric.sphere 0 r := by |
| rintro ⟨z,hz,hzξ⟩ |
| exact hξ ⟨z,hz,(hb z hz).trans hzξ⟩ |
| have hw := meromorphic_boundary_winding_le_zero hh hH hdef hHd hHi hr hr1 ξ hξ' |
| convert hw using 2 |
| ext t |
| change X ((r : ℂ)*circleLoop t)-ξ = H ((r : ℂ)*circleLoop t)-ξ |
| rw [hb _ (by simp [abs_of_pos hr])] |
| have hi := integral_jacobian_nonpos_of_boundary_winding |
| (hXc.of_le (by norm_num)) hr hw |
| have hdet (z) (hz : z ∈ K) : |
| LinearMap.det (fderiv ℝ X z).toLinearMap = ‖deriv h z‖^2-(r^2)⁻¹^2 := by |
| rw [(hX z hz).fderiv_eq] |
| exact det_holomorphic_plus_conjugate |
| ((hh z (hKU hz)).differentiableAt (Metric.isOpen_ball.mem_nhds (hKU hz))) _ |
| have hdcont : ContinuousOn (deriv h) K := |
| (hh.deriv Metric.isOpen_ball).continuousOn.mono hKU |
| have hint : IntegrableOn (fun z => ‖deriv h z‖^2) K := |
| (hdcont.norm.pow 2).integrableOn_compact (isCompact_closedBall 0 r) |
| have hconst : IntegrableOn (fun _ : ℂ => (r^2)⁻¹^2) K := |
| integrableOn_const (isCompact_closedBall 0 r).measure_ne_top |
| have he : (∫ z in K, LinearMap.det (fderiv ℝ X z).toLinearMap) = |
| (∫ z in K, ‖deriv h z‖^2) - volume.real K * (r^2)⁻¹^2 := by |
| rw [setIntegral_congr_fun measurableSet_closedBall hdet, |
| integral_sub hint hconst] |
| simp |
| rw [he] at hi |
| exact sub_nonpos.mp hi |
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| end Brennan |
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| end |
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| end OAI |
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