| import OAI.Geometry.Kahler.BasePotentialControl |
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| namespace OAI |
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| universe uKahler10884_1 uKahler10884_2 uKahler10889_1 uKahler10889_2 uKahler10889_3 |
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|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| namespace PinchedHartogs.BaseConstruction |
| |
| instance : MeasurableSpace Circle := borel Circle |
| instance : BorelSpace Circle := ⟨rfl⟩ |
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|
| def circleMeasure : Measure Circle := |
| Measure.haarMeasure (⟨⟨univ,isCompact_univ⟩,by simp⟩ : TopologicalSpace.PositiveCompacts Circle) |
| |
| instance : IsProbabilityMeasure circleMeasure where |
| measure_univ := Measure.haarMeasure_self |
| instance : Measure.IsHaarMeasure circleMeasure := inferInstanceAs (Measure.IsHaarMeasure (Measure.haarMeasure _)) |
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| def torusMeasure : Measure (Circle × Circle) := circleMeasure.prod circleMeasure |
| instance : IsProbabilityMeasure torusMeasure := inferInstanceAs (IsProbabilityMeasure (circleMeasure.prod circleMeasure)) |
| instance : Measure.IsMulLeftInvariant torusMeasure := inferInstanceAs (Measure.IsMulLeftInvariant (circleMeasure.prod circleMeasure)) |
| instance : Measure.IsMulRightInvariant torusMeasure := inferInstanceAs (Measure.IsMulRightInvariant (circleMeasure.prod circleMeasure)) |
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| def torusAction (z : Circle × Circle) (ξ : Sphere) : Sphere := |
| sphereAction (diagIsometry z.1 z.2 z.1.norm_coe z.2.norm_coe) ξ |
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| @[simp] lemma torusAction_zero (z : Circle × Circle) (ξ : Sphere) : |
| (torusAction z ξ : Base) 0 = (z.1:ℂ)*(ξ:Base) 0 := by simp [torusAction] |
| @[simp] lemma torusAction_one (z : Circle × Circle) (ξ : Sphere) : |
| (torusAction z ξ : Base) 1 = (z.2:ℂ)*(ξ:Base) 1 := by simp [torusAction] |
|
|
| lemma torusAction_continuous : Continuous (fun p : (Circle × Circle) × Sphere => torusAction p.1 p.2) := by |
| apply Continuous.subtype_mk |
| apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℂ)).comp |
| apply continuous_pi |
| intro i |
| fin_cases i |
| · change Continuous (fun p : (Circle × Circle) × Sphere => (torusAction p.1 p.2 : Base) 0) |
| simp_rw [torusAction_zero] |
| fun_prop |
| · change Continuous (fun p : (Circle × Circle) × Sphere => (torusAction p.1 p.2 : Base) 1) |
| simp_rw [torusAction_one] |
| fun_prop |
|
|
| lemma torusAction_mul (z w : Circle × Circle) (ξ : Sphere) : |
| torusAction z (torusAction w ξ) = torusAction (z*w) ξ := by |
| apply Subtype.ext |
| ext i |
| fin_cases i <;> simp [mul_assoc] |
|
|
| def radialSphere (u : unitInterval) : Sphere := |
| ⟨(WithLp.toLp 2 ![(Real.sqrt (u:ℝ):ℂ),(Real.sqrt (1-(u:ℝ)):ℂ)] : Base),by |
| rw [Metric.mem_sphere,dist_zero_right] |
| apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 1)).mp |
| rw [EuclideanSpace.norm_sq_eq] |
| simp only [Fin.sum_univ_two,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one] |
| simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (Real.sqrt_nonneg _),Real.sq_sqrt u.property.1,Real.sq_sqrt (sub_nonneg.mpr u.property.2),one_pow] |
| ring⟩ |
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| @[simp] lemma radialSphere_zero (u : unitInterval) : (radialSphere u : Base) 0 = (Real.sqrt (u:ℝ):ℂ) := rfl |
| @[simp] lemma radialSphere_one (u : unitInterval) : (radialSphere u : Base) 1 = (Real.sqrt (1-(u:ℝ)):ℂ) := rfl |
|
|
| lemma radialSphere_continuous : Continuous radialSphere := by |
| apply Continuous.subtype_mk |
| apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℂ)).comp |
| apply continuous_pi |
| intro i |
| fin_cases i <;> simp only [] <;> fun_prop |
|
|
| attribute [irreducible] torusAction radialSphere |
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| def hopf (p : unitInterval × (Circle × Circle)) : Sphere := torusAction p.2 (radialSphere p.1) |
| lemma hopf_continuous : Continuous hopf := |
| by |
| unfold hopf |
| exact torusAction_continuous.comp (continuous_snd.prodMk (radialSphere_continuous.comp continuous_fst)) |
|
|
| lemma exists_circle_norm_factor (z : ℂ) : ∃ a : Circle, (a:ℂ)*(‖z‖:ℂ) = z := by |
| by_cases hz : z = 0 |
| · subst z; exact ⟨1,by simp⟩ |
| have hn : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz |
| refine ⟨⟨z/(‖z‖:ℂ),?_⟩,?_⟩ |
| · apply mem_sphere_zero_iff_norm.mpr |
| rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (norm_nonneg _),div_self hn] |
| · exact div_mul_cancel₀ z (by exact_mod_cast hn) |
|
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| lemma exists_hopf (ξ : Sphere) : ∃ z : Circle × Circle, torusAction z (radialSphere (radialUnit ξ)) = ξ := by |
| obtain ⟨a,ha⟩ := exists_circle_norm_factor ((ξ:Base) 0) |
| obtain ⟨b,hb⟩ := exists_circle_norm_factor ((ξ:Base) 1) |
| refine ⟨(a,b),?_⟩ |
| apply Subtype.ext |
| ext i |
| fin_cases i |
| · simpa [radialUnit,radialU,Complex.normSq_eq_norm_sq,Real.sqrt_sq (norm_nonneg _)] using ha |
| · have hv : 1-radialU ξ = radialV ξ := by linarith [radialUV ξ] |
| change (torusAction (a,b) (radialSphere (radialUnit ξ)) : Base) 1 = (ξ:Base) 1 |
| rw [torusAction_one,radialSphere_one] |
| change (b:ℂ)*(Real.sqrt (1-radialU ξ):ℂ) = (ξ:Base) 1 |
| rw [hv,radialV,Complex.normSq_eq_norm_sq,Real.sqrt_sq (norm_nonneg _)] |
| exact hb |
|
|
| lemma compact_continuous_integrable {X : Type uKahler10884_1} {E : Type uKahler10884_2} [TopologicalSpace X] [CompactSpace X] |
| [MeasurableSpace X] [OpensMeasurableSpace X] [NormedAddCommGroup E] |
| {μ : Measure X} [IsFiniteMeasure μ] {f : X → E} (hf : Continuous f) : Integrable f μ := |
| hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
|
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| lemma continuous_compact_integral {X : Type uKahler10889_1} {Y : Type uKahler10889_2} {E : Type uKahler10889_3} [TopologicalSpace X] [CompactSpace X] [FirstCountableTopology X] |
| [TopologicalSpace Y] [CompactSpace Y] [MeasurableSpace Y] [OpensMeasurableSpace Y] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [SecondCountableTopology E] {μ : Measure Y} [IsFiniteMeasure μ] |
| {F : X × Y → E} (hF : Continuous F) : Continuous (fun x => ∫ y, F (x,y) ∂μ) := by |
| obtain ⟨C,hC⟩ := isCompact_univ.bddAbove_image hF.norm.continuousOn |
| apply continuous_of_dominated |
| (fun x => (hF.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable) |
| (fun x => Eventually.of_forall (fun y => hC (mem_image_of_mem _ (mem_univ _)))) |
| (integrable_const C) |
| exact Eventually.of_forall (fun y => hF.comp (continuous_id.prodMk continuous_const)) |
|
|
| lemma hopf_integral (f : C(Sphere,ℝ)) : |
| (∫ ξ, f ξ ∂sigma) = ∫ u : unitInterval, ∫ z, f (hopf (u,z)) ∂torusMeasure := by |
| let A (ξ : Sphere) := ∫ z, f (torusAction z ξ) ∂torusMeasure |
| have hc : Continuous A := continuous_compact_integral |
| (f.continuous.comp (torusAction_continuous.comp continuous_swap)) |
| have havg : (∫ ξ, A ξ ∂sigma) = ∫ ξ, f ξ ∂sigma := by |
| rw [show (∫ ξ, A ξ ∂sigma) = ∫ z, ∫ ξ, f (torusAction z ξ) ∂sigma ∂torusMeasure from |
| integral_integral_swap (compact_continuous_integrable (f.continuous.comp (torusAction_continuous.comp continuous_swap)))] |
| simp only [torusAction,sigma_integral_unitary] |
| simp |
| have hA : ∀ ξ, A ξ = A (radialSphere (radialUnit ξ)) := by |
| intro ξ |
| obtain ⟨w,hw⟩ := exists_hopf ξ |
| conv_lhs => rw [← hw] |
| simp only [A,torusAction_mul] |
| exact integral_mul_right_eq_self (fun z => f (torusAction z (radialSphere (radialUnit ξ)))) w |
| rw [← havg] |
| calc |
| _ = ∫ ξ, A (radialSphere (radialUnit ξ)) ∂sigma := integral_congr_ae (Eventually.of_forall hA) |
| _ = _ := radial_continuous_integral ⟨A ∘ radialSphere,hc.comp radialSphere_continuous⟩ |
|
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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