| import OAI.Geometry.KahlerSplitting.MixedShell |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Topology Filter Metric |
| open Set MeasureTheory Metric Filter Topology |
| open scoped ENNReal |
| open scoped ENNReal NNReal |
| open Complex |
| open Set Metric Filter Topology Complex |
| open Set Metric Filter Topology MeasureTheory Complex |
| open Set Metric Filter Topology Asymptotics |
| open scoped Interval |
| open Set Filter Topology Metric |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ENNReal |
| open Set Filter Topology Metric Complex |
| open scoped Real |
| open Set Filter Topology |
| open Set |
| open scoped Manifold ContDiff |
| open Set MeasureTheory |
| open Set Metric |
| open scoped ContDiff |
| open scoped Manifold ENNReal NNReal |
| open Set Metric Filter Topology |
| open scoped Manifold |
|
|
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| lemma IsMixedBox.slice_holomorphic {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W D : Set ℂ} {f : ℂ × ℂ → X.Carrier} |
| (hf : IsMixedBox (S := S) c hc (W ×ˢ D) f) {w : ℂ} (hw : w ∈ W) : |
| MDifferentiableOn 𝓘(ℂ,ℂ) (model n) (fun v ↦ f (w,v)) D := |
| hf.holomorphic.comp |
| (fun z _ ↦ (hasFDerivAt_prodMk_right w z).differentiableAt.mdifferentiableAt.mdifferentiableWithinAt) |
| (fun _ hv ↦ ⟨hw,hv⟩) |
|
|
| lemma IsMixedBox.fibre_eq_of_shell {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {U V : Set ℂ} {f g Fs : ℂ × ℂ → X.Carrier} {a b : ℝ} |
| (ha : 0 ≤ a) (hab : a < b) |
| (hf : IsMixedBox (S := S) c hc (U ×ˢ ball 0 b) f) |
| (hg : IsMixedBox (S := S) c hc (V ×ˢ ball 0 b) g) |
| (hfs : EqOn f Fs (U ×ˢ complexAnnulus a b)) |
| (hgs : EqOn g Fs (V ×ˢ complexAnnulus a b)) |
| {w v : ℂ} (hwU : w ∈ U) (hwV : w ∈ V) (hv : v ∈ ball 0 b) : f (w,v) = g (w,v) := by |
| obtain ⟨z,hz⟩ := complexAnnulus_nonempty ha hab |
| have he : (fun v ↦ f (w,v)) =ᶠ[𝓝 z] (fun v ↦ g (w,v)) := |
| Filter.mem_of_superset ((isOpen_complexAnnulus a b).mem_nhds hz) |
| (fun y hy ↦ (hfs ⟨hwU,hy⟩).trans (hgs ⟨hwV,hy⟩).symm) |
| exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq isOpen_ball |
| (convex_ball (0:ℂ) b).isPreconnected (hf.slice_holomorphic hwU) (hg.slice_holomorphic hwV) |
| (complexAnnulus_subset_ball a b hz) he hv |
|
|
| def MixedFillable (S : HolomorphicSplitting X r s) (c : ℂ × ℂ) (hc : c.1 ≠ c.2) |
| (W : Set ℂ) (Fs : ℂ × ℂ → X.Carrier) (a b : ℝ) : Set ℂ := |
| {w | ∃ e : ℝ, 0 < e ∧ ball w e ⊆ W ∧ ∃ F : ℂ × ℂ → X.Carrier, |
| IsMixedBox (S := S) c hc (ball w e ×ˢ ball 0 b) F ∧ |
| EqOn F Fs (ball w e ×ˢ complexAnnulus a b)} |
|
|
| lemma mixedFillable_subset {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} {Fs : ℂ × ℂ → X.Carrier} {a b : ℝ} : |
| MixedFillable S c hc W Fs a b ⊆ W := by |
| rintro w ⟨e,he,heW,F,hF,heq⟩ |
| exact heW (mem_ball_self he) |
|
|
| lemma ball_subset_mixedFillable {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} {Fs F : ℂ × ℂ → X.Carrier} {a b e : ℝ} {w : ℂ} |
| (hW : ball w e ⊆ W) (hF : IsMixedBox (S := S) c hc (ball w e ×ˢ ball 0 b) F) |
| (heq : EqOn F Fs (ball w e ×ˢ complexAnnulus a b)) : |
| ball w e ⊆ MixedFillable S c hc W Fs a b := by |
| intro x hx |
| obtain ⟨d,hd,hdsub⟩ := Metric.mem_nhds_iff.mp (isOpen_ball.mem_nhds hx) |
| exact ⟨d,hd,hdsub.trans hW,F,hF.mono (prod_mono hdsub subset_rfl), |
| heq.mono (prod_mono hdsub subset_rfl)⟩ |
|
|
| lemma isOpen_mixedFillable {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} {Fs : ℂ × ℂ → X.Carrier} {a b : ℝ} : |
| IsOpen (MixedFillable S c hc W Fs a b) := by |
| apply isOpen_iff_mem_nhds.mpr |
| rintro w ⟨e,he,heW,F,hF,heq⟩ |
| exact Filter.mem_of_superset (ball_mem_nhds w he) (ball_subset_mixedFillable heW hF heq) |
|
|
| theorem glue_all_mixed_fillings {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} {Fs : ℂ × ℂ → X.Carrier} {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) : |
| ∃ F : ℂ × ℂ → X.Carrier, |
| IsMixedBox (S := S) c hc (MixedFillable S c hc W Fs a b ×ˢ ball 0 b) F ∧ |
| EqOn F Fs (MixedFillable S c hc W Fs a b ×ˢ complexAnnulus a b) := by |
| classical |
| let A := MixedFillable S c hc W Fs a b |
| choose e he heW f hf heq using (fun w : A ↦ w.property) |
| let U : A → Set (ℂ × ℂ) := fun w ↦ ball (w:ℂ) (e w) ×ˢ ball 0 b |
| have hU (w : A) : IsOpen (U w) := isOpen_ball.prod isOpen_ball |
| have hcompat (u v : A) : EqOn (f u) (f v) (U u ∩ U v) := by |
| intro z hz |
| exact (hf u).fibre_eq_of_shell ha hab (hf v) (heq u) (heq v) hz.1.1 hz.2.1 hz.1.2 |
| obtain ⟨F,hF,hFeq⟩ := glue_mixed_boxes U hU f hf hcompat (Fs 0) |
| have hUnion : (⋃ w, U w) = A ×ˢ ball 0 b := by |
| ext z |
| constructor |
| · rintro ⟨_,⟨w,rfl⟩,hz⟩ |
| exact ⟨ball_subset_mixedFillable (heW w) (hf w) (heq w) hz.1,hz.2⟩ |
| · intro hz |
| exact mem_iUnion.mpr ⟨⟨z.1,hz.1⟩,mem_ball_self (he ⟨z.1,hz.1⟩),hz.2⟩ |
| refine ⟨F,hUnion ▸ hF,?_⟩ |
| intro z hz |
| let w : A := ⟨z.1,hz.1⟩ |
| exact (hFeq w ⟨mem_ball_self (he w),complexAnnulus_subset_ball _ _ hz.2⟩).trans |
| (heq w ⟨mem_ball_self (he w),hz.2⟩) |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| open scoped Manifold ContDiff ENNReal |
| namespace UniversalCoverSplitting |
| variable {n : ℕ} {X : ComplexManifold n} |
|
|
| lemma first_affine_slope (g : KahlerMetric X) {U : Set (ℂ × ℂ)} (hU : IsOpen U) |
| {f : ℂ × ℂ → X.Carrier} (hf : MDifferentiableOn 𝓘(ℂ,ℂ × ℂ) (model n) f U) |
| (u w : ℂ) (C : ℝ) |
| (hs : ∀ z ∈ U, metricNorm g (f z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) f z (1,0)) ≤ |
| C * metricNorm g (f z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) f z (0,1))) |
| {z : ℂ × ℂ} (hz : separatedAffine (u,0) (w,1) z ∈ U) : |
| metricNorm g ((f ∘ separatedAffine (u,0) (w,1)) z) |
| (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) (f ∘ separatedAffine (u,0) (w,1)) z (1,0)) ≤ |
| (C * ‖w‖) * metricNorm g ((f ∘ separatedAffine (u,0) (w,1)) z) |
| (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) (f ∘ separatedAffine (u,0) (w,1)) z (0,1)) := by |
| have hm := mfderiv_comp z (hf.mdifferentiableAt (hU.mem_nhds hz)) |
| (separatedAffine_hasFDerivAt (u,0) (w,1) z).differentiableAt.mdifferentiableAt |
| rw [hm,mfderiv_eq_fderiv,(separatedAffine_hasFDerivAt (u,0) (w,1) z).fderiv] |
| let D : (ℂ × ℂ) →L[ℂ] Tangent X (f (separatedAffine (u,0) (w,1) z)) := |
| mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) f (separatedAffine (u,0) (w,1) z) |
| change metricNorm g _ (D (separatedLinear (w,1) (1,0))) ≤ |
| (C * ‖w‖) * metricNorm g _ (D (separatedLinear (w,1) (0,1))) |
| have h₁ : separatedLinear (w,1) (1,0) = w • (1,0) := by ext <;> simp [separatedLinear] |
| have h₂ : separatedLinear (w,1) (0,1) = (0,1) := by ext <;> simp [separatedLinear] |
| rw [h₁,h₂,map_smul,metricNorm_complex_smul] |
| calc |
| _ ≤ ‖w‖ * (C * metricNorm g _ (D (0,1))) := mul_le_mul_of_nonneg_left (hs _ hz) (norm_nonneg _) |
| _ = _ := by ring |
|
|
| theorem radial_disk_area_bound (g : KahlerMetric X) |
| {A : Set ℂ} {u : ℂ} (hA : IsOpen A) {f : ℂ × ℂ → X.Carrier} {b ρ C R R₀ : ℝ} |
| (hf : MDifferentiableOn 𝓘(ℂ,ℂ × ℂ) (model n) f (A ×ˢ ball 0 b)) |
| (_hρ : 0 < ρ) (hρA : ball u ρ ⊆ A) (hC : 0 ≤ C) |
| (hR : 0 < R) (hRR : R + C * ρ < R₀) (hRb : R₀ < b) |
| (hs : ∀ z ∈ A ×ˢ ball 0 b, |
| metricNorm g (f z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) f z (1,0)) ≤ |
| C * metricNorm g (f z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) f z (0,1))) : |
| ∀ w ∈ ball u ρ, curveArea g (fun v ↦ f (w,v)) (closedBall 0 R) ≤ |
| curveArea g (fun v ↦ f (u,v)) (ball 0 R₀) := by |
| intro w hw |
| have hw' : ‖w-u‖ < ρ := by simpa [dist_eq_norm] using hw |
| let T := separatedAffine (u,0) (w-u,1) |
| let U := T ⁻¹' (A ×ˢ ball 0 b) |
| let D := C * ‖w-u‖ |
| let R' : ℝ → ℝ := fun x ↦ R₀ - D * x |
| have hD : 0 ≤ D := mul_nonneg hC (norm_nonneg _) |
| have hmargin : R < R₀ - D := by |
| have := mul_le_mul_of_nonneg_left hw'.le hC |
| dsimp [D] |
| linarith |
| have hT : Continuous T := continuous_iff_continuousAt.mpr fun z ↦ (separatedAffine_hasFDerivAt (u,0) (w-u,1) z).continuousAt |
| have hU : IsOpen U := (hA.prod isOpen_ball).preimage hT |
| have hf' : MDifferentiableOn 𝓘(ℂ,ℂ × ℂ) (model n) (f ∘ T) U := |
| hf.comp (fun z _ ↦ (separatedAffine_hasFDerivAt (u,0) (w-u,1) z).differentiableAt.mdifferentiableAt.mdifferentiableWithinAt) |
| (fun _ hz ↦ hz) |
| have hRp (x : ℝ) (hx : x ∈ Icc (0:ℝ) 1) : 0 ≤ R' x := by |
| have := mul_le_mul_of_nonneg_left hx.2 hD |
| dsimp [R'] |
| linarith |
| have hslices : ∀ x ∈ Icc (0:ℝ) 1, MapsTo (fun v : ℂ ↦ ((x:ℂ),v)) (closedBall 0 (R' x)) U := by |
| intro x hx v hv |
| change separatedAffine (u,0) (w-u,1) ((x:ℂ),v) ∈ A ×ˢ ball 0 b |
| simp only [separatedAffine,zero_add,one_mul] |
| constructor |
| · apply hρA |
| simp only [mem_ball,dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hx.1] |
| exact (mul_le_of_le_one_right (norm_nonneg (w-u)) hx.2).trans_lt hw' |
| · have hv' : ‖v‖ ≤ R₀ - D*x := by simpa [R'] using hv |
| have := mul_nonneg hD hx.1 |
| simpa only [mem_ball,dist_zero_right] using (show ‖v‖ < b by linarith) |
| have hm := moving_disk_area_le_of_slices g hU isOpen_univ hf' |
| (show ContDiffOn ℝ ∞ R' univ from (by fun_prop : ContDiff ℝ ∞ R').contDiffOn) |
| (show (0:ℝ) ≤ 1 by norm_num) (subset_univ _) hRp |
| (fun x _ ↦ by |
| have hd : HasDerivAt R' (-D) x := by simpa [R'] using ((hasDerivAt_id x).const_mul D).const_sub R₀ |
| exact hd.deriv.le) hslices |
| (fun z hz ↦ first_affine_slope g (hA.prod isOpen_ball) hf u (w-u) C hs hz) |
| have hfinal : curveArea g (fun v ↦ f (w,v)) (ball 0 (R₀-D)) ≤ |
| curveArea g (fun v ↦ f (u,v)) (ball 0 R₀) := by |
| simpa only [Function.comp_apply,T,separatedAffine,R',mul_one,mul_zero,sub_zero,zero_add,one_mul,Complex.ofReal_one,Complex.ofReal_zero,add_sub_cancel,add_zero] using hm |
| exact (curveArea_mono g _ (closedBall_subset_ball hmargin)).trans hfinal |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology |
| namespace UniversalCoverSplitting |
| |
| theorem ball_subset_of_closedBall_continuation {E : Type*} [MetricSpace E] [ProperSpace E] |
| {A : Set E} (hA : IsOpen A) {x : E} (hx : x ∈ A) {e : ℝ} |
| (hfill : ∀ ρ : ℝ, 0 < ρ → ρ < e → ball x ρ ⊆ A → closedBall x ρ ⊆ A) : |
| ball x e ⊆ A := by |
| by_contra h |
| obtain ⟨y,hy,hny⟩ := not_subset.mp h |
| obtain ⟨z,hz,hd⟩ := hA.isClosed_compl.exists_infDist_eq_dist ⟨y,hny⟩ x |
| have hpos : 0 < infDist x Aᶜ := by |
| rw [hd] |
| exact dist_pos.mpr (ne_of_mem_of_not_mem hx hz) |
| have hlt : infDist x Aᶜ < e := |
| (infDist_le_dist_of_mem hny).trans_lt (mem_ball'.mp hy) |
| apply hz |
| apply hfill _ hpos hlt ball_infDist_compl_subset |
| rw [mem_closedBall,dist_comm,← hd] |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| open scoped Manifold ENNReal NNReal |
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| lemma mixedFillable_closedBall [CompactSpace X.Carrier] |
| (g : KahlerMetric X) (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} (hW : IsOpen W) {a b R R₀ ρ C : ℝ} (ha : 0 ≤ a) |
| (haR : a < R) (hRR : R + C*ρ < R₀) (hRb : R₀ < b) |
| (hρ : 0 < ρ) (hC : 0 ≤ C) |
| {Fs : ℂ × ℂ → X.Carrier} (hs : IsMixedBox (S := S) c hc (W ×ˢ complexAnnulus a b) Fs) |
| {F : ℂ × ℂ → X.Carrier} |
| (hF : IsMixedBox (S := S) c hc (MixedFillable S c hc W Fs a b ×ˢ ball 0 b) F) |
| (hFs : EqOn F Fs (MixedFillable S c hc W Fs a b ×ˢ complexAnnulus a b)) |
| (hslope : ∀ z ∈ MixedFillable S c hc W Fs a b ×ˢ ball 0 b, |
| metricNorm g (F z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) F z (1,0)) ≤ |
| C * metricNorm g (F z) (mfderiv 𝓘(ℂ,ℂ × ℂ) (model n) F z (0,1))) |
| {u : ℂ} (hball : ball u ρ ⊆ MixedFillable S c hc W Fs a b) |
| (hclosed : closedBall u ρ ⊆ W) : |
| closedBall u ρ ⊆ MixedFillable S c hc W Fs a b := by |
| have hR : 0 < R := ha.trans_lt haR |
| have hRR' : R < R₀ := lt_of_le_of_lt (le_add_of_nonneg_right (mul_nonneg hC hρ.le)) hRR |
| have hu : u ∈ MixedFillable S c hc W Fs a b := hball (mem_ball_self hρ) |
| have hfin : curveArea g (fun v ↦ F (u,v)) (ball 0 R₀) ≠ (⊤ : ℝ≥0∞) := |
| curveArea_slice_ne_top g (isOpen_mixedFillable.prod isOpen_ball) hF.holomorphic |
| (fun v hv ↦ ⟨hu,closedBall_subset_ball hRb hv⟩) |
| let B : ℝ≥0 := (curveArea g (fun v ↦ F (u,v)) (ball 0 R₀)).toNNReal |
| have hbound := radial_disk_area_bound g isOpen_mixedFillable hF.holomorphic hρ hball hC hR hRR hRb hslope |
| intro w hw |
| have hw' : w ∈ closure (ball u ρ) := by rwa [closure_ball u hρ.ne'] |
| obtain ⟨wj,hwj,hlim⟩ := mem_closure_iff_seq_limit.mp hw' |
| exact mixed_shell_fill_of_bounded_slices g h₁ h₂ hW (hclosed hw) ha haR (hRR'.trans hRb) hs |
| wj (fun j ↦ mixedFillable_subset (hball (hwj j))) hlim |
| (fun j v ↦ F (wj j,v)) (fun j ↦ hF.slice_holomorphic (hball (hwj j))) |
| (fun j v hv ↦ hFs ⟨hball (hwj j),hv⟩) B |
| (fun j ↦ by simpa only [B,ENNReal.coe_toNNReal hfin] using hbound _ (hwj j)) |
| |
| theorem mixedFillable_uniform_local_continuation [CompactSpace X.Carrier] |
| (g : KahlerMetric X) (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) (c : ℂ × ℂ) (hc : c.1 ≠ c.2) |
| {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ (W : Set ℂ) (Fs : ℂ × ℂ → X.Carrier), |
| IsOpen W → IsMixedBox (S := S) c hc (W ×ˢ complexAnnulus a b) Fs → |
| ∀ u ∈ MixedFillable S c hc W Fs a b, ∀ e : ℝ, 0 < e → e ≤ δ → |
| ball u e ⊆ W → ball u e ⊆ MixedFillable S c hc W Fs a b := by |
| obtain ⟨C,hC,hCs⟩ := exists_uniform_mixed_slope g S c hc |
| obtain ⟨R,haR,hRb⟩ := exists_between hab |
| obtain ⟨R₀,hRR,hR₀b⟩ := exists_between hRb |
| let δ := (R₀-R)/(C+1) |
| have hCp : 0 < C+1 := by positivity |
| have hδ : 0 < δ := div_pos (sub_pos.mpr hRR) hCp |
| refine ⟨δ,hδ,?_⟩ |
| intro W Fs hW hs u hu e he heδ heW |
| obtain ⟨F,hF,hFs⟩ := glue_all_mixed_fillings (S := S) (W := W) (Fs := Fs) ha hab |
| apply ball_subset_of_closedBall_continuation isOpen_mixedFillable hu |
| intro ρ hρ hρe hρA |
| have hρδ : ρ < δ := hρe.trans_le heδ |
| have hmargin : R + C*ρ < R₀ := by |
| have hh : ρ*(C+1) < R₀-R := (lt_div_iff₀ hCp).mp hρδ |
| have hρpos := hρ |
| nlinarith |
| exact mixedFillable_closedBall g h₁ h₂ hW ha haR hmargin hR₀b hρ hC hs hF hFs |
| (hCs _ _ (isOpen_mixedFillable.prod isOpen_ball) hF) hρA |
| ((closedBall_subset_ball hρe).trans heW) |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology |
| namespace UniversalCoverSplitting |
| |
| theorem subset_of_uniform_local_continuation {E : Type*} [MetricSpace E] |
| {W A : Set E} (hW : IsOpen W) (hWc : IsPreconnected W) (hA : IsOpen A) |
| (hAW : A ⊆ W) (hAn : A.Nonempty) {δ : ℝ} (hδ : 0 < δ) |
| (hcont : ∀ u ∈ A, ∀ e : ℝ, 0 < e → e ≤ δ → ball u e ⊆ W → ball u e ⊆ A) : |
| W ⊆ A := by |
| apply hWc.subset_of_closure_inter_subset hA |
| · obtain ⟨u,hu⟩ := hAn |
| exact ⟨u,hAW hu,hu⟩ |
| intro x hx |
| obtain ⟨ε,hε,hεW⟩ := Metric.mem_nhds_iff.mp (hW.mem_nhds hx.2) |
| let d := min (ε/4) (δ/4) |
| have hd : 0 < d := lt_min (by positivity) (by positivity) |
| have hdε : d ≤ ε/4 := min_le_left _ _ |
| have hdδ : d ≤ δ/4 := min_le_right _ _ |
| obtain ⟨u,hux,huA⟩ := mem_closure_iff.mp hx.1 (ball x d) isOpen_ball (mem_ball_self hd) |
| have hux' : dist u x < d := hux |
| have hballW : ball u (2*d) ⊆ W := by |
| intro y hy |
| apply hεW |
| have hy' : dist y u < 2*d := hy |
| have ht := dist_triangle y u x |
| change dist y x < ε |
| linarith |
| apply hcont u huA (2*d) (by positivity) (by linarith) hballW |
| change dist x u < 2*d |
| rw [dist_comm] |
| linarith |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology |
| open scoped Manifold |
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
| |
| theorem mixed_shell_extension [CompactSpace X.Carrier] |
| (g : KahlerMetric X) (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W : Set ℂ} (hW : IsOpen W) (hWc : IsPreconnected W) {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) |
| {Fs : ℂ × ℂ → X.Carrier} (hs : IsMixedBox (S := S) c hc (W ×ˢ complexAnnulus a b) Fs) |
| {w : ℂ} (hw : w ∈ W) {H : ℂ → X.Carrier} |
| (hH : MDifferentiableOn 𝓘(ℂ,ℂ) (model n) H (ball 0 b)) |
| (hHs : ∀ v ∈ complexAnnulus a b, H v = Fs (w,v)) : |
| ∃ F : ℂ × ℂ → X.Carrier, IsMixedBox (S := S) c hc (W ×ˢ ball 0 b) F ∧ |
| EqOn F Fs (W ×ˢ complexAnnulus a b) := by |
| have hwA : w ∈ MixedFillable S c hc W Fs a b := |
| mixed_shell_fill_of_slice h₁ h₂ hW hw ha hab le_rfl hs hH hHs |
| obtain ⟨δ,hδ,hcont⟩ := mixedFillable_uniform_local_continuation g h₁ h₂ c hc ha hab |
| have hWA : W ⊆ MixedFillable S c hc W Fs a b := |
| subset_of_uniform_local_continuation hW hWc isOpen_mixedFillable mixedFillable_subset |
| ⟨w,hwA⟩ hδ (hcont W Fs hW hs) |
| have hAW : MixedFillable S c hc W Fs a b = W := Subset.antisymm mixedFillable_subset hWA |
| obtain ⟨F,hF,hFs⟩ := glue_all_mixed_fillings (S := S) (c := c) (hc := hc) (W := W) (Fs := Fs) ha hab |
| rw [hAW] at hF hFs |
| exact ⟨F,hF,hFs⟩ |
| |
| theorem mixed_hartogs_extension [CompactSpace X.Carrier] |
| (g : KahlerMetric X) (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {c : ℂ × ℂ} {hc : c.1 ≠ c.2} |
| {W U : Set ℂ} (hW : IsOpen W) (hWc : IsPreconnected W) (hUW : U ⊆ W) (hUn : U.Nonempty) |
| {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) {f : ℂ × ℂ → X.Carrier} |
| (hf : IsMixedBox (S := S) c hc ((U ×ˢ ball 0 b) ∪ (W ×ˢ complexAnnulus a b)) f) : |
| ∃ F : ℂ × ℂ → X.Carrier, IsMixedBox (S := S) c hc (W ×ˢ ball 0 b) F ∧ |
| EqOn F f ((U ×ˢ ball 0 b) ∪ (W ×ˢ complexAnnulus a b)) ∧ |
| ∀ G : ℂ × ℂ → X.Carrier, IsMixedBox (S := S) c hc (W ×ˢ ball 0 b) G → |
| EqOn G f (W ×ˢ complexAnnulus a b) → EqOn G F (W ×ˢ ball 0 b) := by |
| obtain ⟨w,hw⟩ := hUn |
| have hcap : IsMixedBox (S := S) c hc (U ×ˢ ball 0 b) f := hf.mono subset_union_left |
| have hs : IsMixedBox (S := S) c hc (W ×ˢ complexAnnulus a b) f := hf.mono subset_union_right |
| obtain ⟨F,hF,hFs⟩ := mixed_shell_extension g h₁ h₂ hW hWc ha hab hs (hUW hw) |
| (hcap.slice_holomorphic hw) (fun _ _ ↦ rfl) |
| refine ⟨F,hF,?_,?_⟩ |
| · intro z hz |
| rcases hz with hz|hz |
| · exact hF.fibre_eq_of_shell ha hab hcap hFs (fun _ _ ↦ rfl) (hUW hz.1) hz.1 hz.2 |
| · exact hFs hz |
| · intro G hG hGs z hz |
| exact hG.fibre_eq_of_shell ha hab hF hGs hFs hz.1 hz.1 hz.2 |
| |
| end UniversalCoverSplitting |
| |
| end |
| |
| end OAI |
| |
