| import OAI.Geometry.KahlerSplitting.Superellipsoids |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Topology Filter Metric |
| open Set MeasureTheory Metric Filter Topology |
| 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 Filter Topology Metric Complex |
| open scoped Real |
| open Set Filter Topology |
| open scoped Manifold ContDiff |
| open Set MeasureTheory |
| open scoped ContDiff |
| open scoped Manifold ContDiff ENNReal |
| open scoped Manifold ENNReal NNReal |
| open Set Metric |
| open Set MeasureTheory Filter Topology |
| open scoped ENNReal |
| open Set |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ENNReal |
| open Set Metric Filter Topology |
| open scoped Manifold |
|
|
| namespace UniversalCoverSplitting |
| variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s} |
|
|
| theorem box_axes_germ_unique (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) {U V : Set ℂ} (hU : IsOpen U) (hV : IsOpen V) |
| {f g : ℂ × ℂ → X.Carrier} (hf : IsBox S (U ×ˢ V) f) (hg : IsBox S (U ×ˢ V) g) |
| (p q : ℂ) (hp : p ∈ U) (hq : q ∈ V) |
| (heq₁ : ∀ z ∈ U, f (z,q) = g (z,q)) (heq₂ : ∀ z ∈ V, f (p,z) = g (p,z)) : |
| f =ᶠ[𝓝 (p,q)] g := by |
| obtain ⟨e,het,_⟩ := exists_splittingChart S h₁ h₂ (f (p,q)) |
| have hpp : (p,q) ∈ U ×ˢ V := ⟨hp,hq⟩ |
| have hfn := (hf.1.continuousOn.continuousAt ((hU.prod hV).mem_nhds hpp)) |
| (e.chart.open_source.mem_nhds het) |
| have hgn := (hg.1.continuousOn.continuousAt ((hU.prod hV).mem_nhds hpp)) |
| (e.chart.open_source.mem_nhds ((heq₁ p hp) ▸ het)) |
| obtain ⟨a,ha,hasub⟩ := Metric.mem_nhds_iff.mp (inter_mem ((hU.prod hV).mem_nhds hpp) (inter_mem hfn hgn)) |
| have hprod : ball p a ×ˢ ball q a = ball (p,q) a := ball_prod_same _ _ _ |
| have hsub₁ : ball p a ⊆ U := by |
| intro z hz |
| exact (hasub (hprod ▸ (show (z,q) ∈ ball p a ×ˢ ball q a from ⟨hz,mem_ball_self ha⟩))).1.1 |
| have hsub₂ : ball q a ⊆ V := by |
| intro z hz |
| exact (hasub (hprod ▸ (show (p,z) ∈ ball p a ×ˢ ball q a from ⟨mem_ball_self ha,hz⟩))).1.2 |
| have hm (z : ℂ × ℂ) (hz : z ∈ ball p a ×ˢ ball q a) : |
| f z ∈ e.chart.source ∧ g z ∈ e.chart.source := (hasub (hprod ▸ hz)).2 |
| have hbsub : ball p a ×ˢ ball q a ⊆ U ×ˢ V := prod_mono hsub₁ hsub₂ |
| have hfb := e.box_coordinates (isOpen_ball.prod isOpen_ball) |
| (show IsBox S (ball p a ×ˢ ball q a) f from ⟨hf.1.mono hbsub,fun z hz ↦ hf.2 z (hbsub hz)⟩) |
| (fun z hz ↦ (hm z hz).1) |
| have hgb := e.box_coordinates (isOpen_ball.prod isOpen_ball) |
| (show IsBox S (ball p a ×ˢ ball q a) g from ⟨hg.1.mono hbsub,fun z hz ↦ hg.2 z (hbsub hz)⟩) |
| (fun z hz ↦ (hm z hz).2) |
| filter_upwards [ball_mem_nhds (p,q) ha] with z hz |
| have hzz : z ∈ ball p a ×ˢ ball q a := hprod.symm ▸ hz |
| apply e.chart.injOn (hm z hzz).1 (hm z hzz).2 |
| have hfe := separated_eq_axes isOpen_ball isOpen_ball (convex_ball p a).isPreconnected |
| (convex_ball q a).isPreconnected hfb.1 hfb.2 hzz.1 (mem_ball_self ha) hzz.2 (mem_ball_self ha) |
| have hge := separated_eq_axes isOpen_ball isOpen_ball (convex_ball p a).isPreconnected |
| (convex_ball q a).isPreconnected hgb.1 hgb.2 hzz.1 (mem_ball_self ha) hzz.2 (mem_ball_self ha) |
| change (e.chart ∘ f) z = (e.chart ∘ g) z |
| rw [hfe,hge] |
| simp only [Function.comp_apply,heq₁ z.1 (hsub₁ hzz.1),heq₂ z.2 (hsub₂ hzz.2)] |
|
|
| 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} |
|
|
| lemma horizontal_axis_boundary_extension (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) |
| {U : Set (ℂ × ℂ)} (hU : IsOpen U) (hUc : Convex ℝ U) |
| {f : ℂ × ℂ → X.Carrier} (hf : IsBox S U f) |
| {O : Set ℂ} (hO : IsOpen O) {α : ℂ → X.Carrier} (hα : IsHorizontalCurve S O α) |
| {p : ℂ} (hp : p ∈ O) (hpc : p ∈ closure {z : ℂ | (z,0) ∈ U}) |
| (haxis : ∀ z ∈ O, (z,0) ∈ U → f (z,0) = α z) : |
| ∃ V : Set (ℂ × ℂ), IsOpen V ∧ (p,0) ∈ V ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S V F ∧ EqOn F f (V ∩ U) := by |
| obtain ⟨e,hep,_⟩ := exists_splittingChart S h₁ h₂ (α p) |
| have hαn := (hα.1.continuousOn.continuousAt (hO.mem_nhds hp)) (e.chart.open_source.mem_nhds hep) |
| obtain ⟨d,hd,hdO⟩ := Metric.mem_nhds_iff.mp (inter_mem (hO.mem_nhds hp) hαn) |
| have hdsub : ball p d ⊆ O := fun _ hz ↦ (hdO hz).1 |
| have hdae : MapsTo α (ball p d) e.chart.source := fun _ hz ↦ (hdO hz).2 |
| obtain ⟨q,hq,hqp⟩ := mem_closure_iff_nhds.mp hpc (ball p d) (ball_mem_nhds p hd) |
| have hqe : f (q,0) ∈ e.chart.source := (haxis q (hdsub hq) hqp) ▸ hdae hq |
| have hfn := (hf.1.continuousOn.continuousAt (hU.mem_nhds hqp)) (e.chart.open_source.mem_nhds hqe) |
| obtain ⟨a,ha,haV⟩ := Metric.mem_nhds_iff.mp (inter_mem (hU.mem_nhds hqp) hfn) |
| have hβmap (z : ℂ) (hz : z ∈ ball (0:ℂ) a) : (q,z) ∈ U ∧ f (q,z) ∈ e.chart.source := by |
| apply haV |
| rw [← ball_prod_same] |
| exact ⟨mem_ball_self ha,hz⟩ |
| let β : ℂ → X.Carrier := fun z ↦ f (q,z) |
| have hβ : IsVerticalCurve S (ball 0 a) β := hf.vertical_slice hU q (fun z hz ↦ (hβmap z hz).1) |
| let V := ball p d ×ˢ ball (0:ℂ) a |
| let F : ℂ × ℂ → X.Carrier := fun z ↦ e.chart.symm ((e.chart (α z.1)).1,(e.chart (β z.2)).2) |
| have hF : IsBox S V F := e.mix_curves isOpen_ball isOpen_ball (hα.1.mono hdsub) |
| hβ.1 hdae (fun z hz ↦ (hβmap z hz).2) |
| have haxes := e.mix_curves_axes isOpen_ball isOpen_ball (convex_ball p d).isPreconnected |
| (convex_ball (0:ℂ) a).isPreconnected (hα.mono hdsub) hβ hdae (fun z hz ↦ (hβmap z hz).2) |
| hq (mem_ball_self ha) (haxis q (hdsub hq) hqp).symm |
| have hVo : IsOpen V := isOpen_ball.prod isOpen_ball |
| have hqc : (q,0) ∈ V ∩ U := ⟨⟨hq,mem_ball_self ha⟩,hqp⟩ |
| obtain ⟨b,hb,hbV⟩ := Metric.mem_nhds_iff.mp ((hVo.inter hU).mem_nhds hqc) |
| have hbsub : ball q b ×ˢ ball (0:ℂ) b ⊆ V ∩ U := by rwa [ball_prod_same] |
| have hFg : F =ᶠ[𝓝 (q,0)] f := by |
| apply box_axes_germ_unique h₁ h₂ isOpen_ball isOpen_ball |
| (hF.mono (hbsub.trans inter_subset_left)) (hf.mono (hbsub.trans inter_subset_right)) q 0 |
| (mem_ball_self hb) (mem_ball_self hb) |
| · intro z hz |
| have hm := hbsub (show (z,0) ∈ ball q b ×ˢ ball (0:ℂ) b from ⟨hz,mem_ball_self hb⟩) |
| exact (haxes.1 z hm.1.1).trans (haxis z (hdsub hm.1.1) hm.2).symm |
| · intro z hz |
| have hm := hbsub (show (q,z) ∈ ball q b ×ˢ ball (0:ℂ) b from ⟨mem_ball_self hb,hz⟩) |
| exact haxes.2 z hm.1.2 |
| refine ⟨V,hVo,⟨mem_ball_self hd,mem_ball_self ha⟩,F,hF,?_⟩ |
| exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq (hVo.inter hU) |
| (((convex_ball p d).prod (convex_ball (0:ℂ) a)).inter hUc).isPreconnected |
| (hF.1.mono inter_subset_left) (hf.1.mono inter_subset_right) hqc hFg |
|
|
| lemma vertical_axis_boundary_extension (h₁ : Integrable S.projection) |
| (h₂ : Integrable (complementaryProjection S)) |
| {U : Set (ℂ × ℂ)} (hU : IsOpen U) (hUc : Convex ℝ U) |
| {f : ℂ × ℂ → X.Carrier} (hf : IsBox S U f) |
| {O : Set ℂ} (hO : IsOpen O) {α : ℂ → X.Carrier} (hα : IsVerticalCurve S O α) |
| {p : ℂ} (hp : p ∈ O) (hpc : p ∈ closure {z : ℂ | (0,z) ∈ U}) |
| (haxis : ∀ z ∈ O, (0,z) ∈ U → f (0,z) = α z) : |
| ∃ V : Set (ℂ × ℂ), IsOpen V ∧ (0,p) ∈ V ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S V F ∧ EqOn F f (V ∩ U) := by |
| obtain ⟨e,hep,_⟩ := exists_splittingChart S h₁ h₂ (α p) |
| have hαn := (hα.1.continuousOn.continuousAt (hO.mem_nhds hp)) (e.chart.open_source.mem_nhds hep) |
| obtain ⟨d,hd,hdO⟩ := Metric.mem_nhds_iff.mp (inter_mem (hO.mem_nhds hp) hαn) |
| have hdsub : ball p d ⊆ O := fun _ hz ↦ (hdO hz).1 |
| have hdae : MapsTo α (ball p d) e.chart.source := fun _ hz ↦ (hdO hz).2 |
| obtain ⟨q,hq,hqp⟩ := mem_closure_iff_nhds.mp hpc (ball p d) (ball_mem_nhds p hd) |
| have hqe : f (0,q) ∈ e.chart.source := (haxis q (hdsub hq) hqp) ▸ hdae hq |
| have hfn := (hf.1.continuousOn.continuousAt (hU.mem_nhds hqp)) (e.chart.open_source.mem_nhds hqe) |
| obtain ⟨a,ha,haV⟩ := Metric.mem_nhds_iff.mp (inter_mem (hU.mem_nhds hqp) hfn) |
| have hβmap (z : ℂ) (hz : z ∈ ball (0:ℂ) a) : (z,q) ∈ U ∧ f (z,q) ∈ e.chart.source := by |
| apply haV |
| rw [← ball_prod_same] |
| exact ⟨hz,mem_ball_self ha⟩ |
| let β : ℂ → X.Carrier := fun z ↦ f (z,q) |
| have hβ : IsHorizontalCurve S (ball 0 a) β := hf.horizontal_slice hU q (fun z hz ↦ (hβmap z hz).1) |
| let V := ball (0:ℂ) a ×ˢ ball p d |
| let F : ℂ × ℂ → X.Carrier := fun z ↦ e.chart.symm ((e.chart (β z.1)).1,(e.chart (α z.2)).2) |
| have hF : IsBox S V F := e.mix_curves isOpen_ball isOpen_ball hβ.1 (hα.1.mono hdsub) |
| (fun z hz ↦ (hβmap z hz).2) hdae |
| have haxes := e.mix_curves_axes isOpen_ball isOpen_ball (convex_ball (0:ℂ) a).isPreconnected |
| (convex_ball p d).isPreconnected hβ (hα.mono hdsub) (fun z hz ↦ (hβmap z hz).2) hdae |
| (mem_ball_self ha) hq (haxis q (hdsub hq) hqp) |
| have hVo : IsOpen V := isOpen_ball.prod isOpen_ball |
| have hqc : (0,q) ∈ V ∩ U := ⟨⟨mem_ball_self ha,hq⟩,hqp⟩ |
| obtain ⟨b,hb,hbV⟩ := Metric.mem_nhds_iff.mp ((hVo.inter hU).mem_nhds hqc) |
| have hbsub : ball (0:ℂ) b ×ˢ ball q b ⊆ V ∩ U := by rwa [ball_prod_same] |
| have hFg : F =ᶠ[𝓝 (0,q)] f := by |
| apply box_axes_germ_unique h₁ h₂ isOpen_ball isOpen_ball |
| (hF.mono (hbsub.trans inter_subset_left)) (hf.mono (hbsub.trans inter_subset_right)) 0 q |
| (mem_ball_self hb) (mem_ball_self hb) |
| · intro z hz |
| have hm := hbsub (show (z,q) ∈ ball (0:ℂ) b ×ˢ ball q b from ⟨hz,mem_ball_self hb⟩) |
| exact haxes.1 z hm.1.1 |
| · intro z hz |
| have hm := hbsub (show (0,z) ∈ ball (0:ℂ) b ×ˢ ball q b from ⟨mem_ball_self hb,hz⟩) |
| exact (haxes.2 z hm.1.2).trans (haxis z (hdsub hm.1.2) hm.2).symm |
| refine ⟨V,hVo,⟨mem_ball_self ha,mem_ball_self hd⟩,F,hF,?_⟩ |
| exact holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq (hVo.inter hU) |
| (((convex_ball (0:ℂ) a).prod (convex_ball p d)).inter hUc).isPreconnected |
| (hF.1.mono inter_subset_left) (hf.1.mono inter_subset_right) hqc hFg |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology |
| namespace UniversalCoverSplitting |
|
|
| lemma superPotential_norm_bound {m : ℕ} (hm : 0 < m) {z : ℂ × ℂ} {a : ℝ} |
| (hz : superPotential m z ≤ a) : ‖z‖ ≤ max a 1 + 1 := by |
| have hb (w : ℂ) (hw : Complex.normSq w ^ m ≤ a) : ‖w‖ ≤ max a 1 + 1 := by |
| by_cases h : ‖w‖ ≤ 1 |
| · linarith [le_max_right a 1] |
| have hh : 1 ≤ Complex.normSq w := by rw [Complex.normSq_eq_norm_sq]; nlinarith [norm_nonneg w] |
| have hpow := le_self_pow₀ hh hm.ne' |
| rw [Complex.normSq_eq_norm_sq] at hpow hw |
| nlinarith [le_max_left a 1,le_max_right a 1,norm_nonneg w] |
| exact max_le (hb z.1 (le_trans (le_add_of_nonneg_right |
| (pow_nonneg (Complex.normSq_nonneg z.2) m)) hz)) |
| (hb z.2 (le_trans (le_add_of_nonneg_left |
| (pow_nonneg (Complex.normSq_nonneg z.1) m)) hz)) |
| |
| lemma superSublevel_compact {m : ℕ} (hm : 0 < m) (a : ℝ) : |
| IsCompact {z | superPotential m z ≤ a} := by |
| apply isCompact_iff_isClosed_bounded.mpr |
| refine ⟨isClosed_le (superPotential_contDiff m (k := 0)).continuous continuous_const,?_⟩ |
| apply (isBounded_closedBall (x := (0:ℂ × ℂ)) (r := max a 1+1)).subset |
| intro z hz |
| rw [mem_closedBall,dist_zero_right] |
| exact superPotential_norm_bound hm hz |
| |
| lemma superPotential_radial_lt {m : ℕ} (hm : 0 < m) {a t : ℝ} (ha : 0 < a) |
| (ht : 0 ≤ t) (ht1 : t < 1) {z : ℂ × ℂ} (hz : superPotential m z ≤ a) : |
| t • z ∈ superDomain m a := by |
| change superPotential m (t • z) < a |
| rw [superPotential_smul] |
| have htP : t^(2*m) < 1 := pow_lt_one₀ ht ht1 (by omega) |
| exact lt_of_le_of_lt (mul_le_mul_of_nonneg_left hz (pow_nonneg ht _)) (by nlinarith) |
| |
| lemma superDomain_closure {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : 0 < a) : |
| closure (superDomain m a) = {z | superPotential m z ≤ a} := by |
| apply Subset.antisymm |
| · exact closure_minimal (fun z (h : superPotential m z < a) ↦ h.le) |
| (isClosed_le (superPotential_contDiff m (k := 0)).continuous continuous_const) |
| intro z hz |
| have ht : Tendsto (fun t : ℝ ↦ t • z) (𝓝[<] 1) (𝓝 z) := by |
| have hc : Continuous (fun t : ℝ ↦ t • (z : ℂ × ℂ)) := continuous_id.smul continuous_const |
| have hh := hc.continuousAt (x := (1:ℝ)) |>.tendsto |
| rw [one_smul] at hh |
| exact hh.mono_left nhdsWithin_le_nhds |
| apply mem_closure_of_tendsto ht |
| filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds (show (0:ℝ)<1 by norm_num))] |
| with t ht ht0 |
| exact superPotential_radial_lt hm ha ht0.le ht hz |
| |
| lemma superDomain_frontier {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : 0 < a) : |
| frontier (superDomain m a) = {z | superPotential m z = a} := by |
| rw [frontier, (superDomain_open m a).interior_eq, superDomain_closure hm ha] |
| ext z |
| change (superPotential m z ≤ a ∧ ¬superPotential m z < a) ↔ superPotential m z = a |
| rw [not_lt] |
| exact ⟨fun h ↦ le_antisymm h.1 h.2, fun h ↦ ⟨h.le,h.ge⟩⟩ |
| |
| lemma superDomain_axis_closure {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : 0 < a) |
| {p : ℂ} (hp : superPotential m (p,0) ≤ a) : |
| p ∈ closure {z : ℂ | (z,0) ∈ superDomain m a} := by |
| have ht : Tendsto (fun t : ℝ ↦ t • p) (𝓝[<] 1) (𝓝 p) := by |
| have hc : Continuous (fun t : ℝ ↦ t • (p : ℂ)) := continuous_id.smul continuous_const |
| have hh := hc.continuousAt (x := (1:ℝ)) |>.tendsto |
| rw [one_smul] at hh |
| exact hh.mono_left nhdsWithin_le_nhds |
| apply mem_closure_of_tendsto ht |
| filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds (show (0:ℝ)<1 by norm_num))] |
| with t ht ht0 |
| simpa only [Prod.smul_mk,smul_zero] using superPotential_radial_lt hm ha ht0.le ht hp |
| |
| lemma superDomain_vertical_axis_closure {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : 0 < a) |
| {p : ℂ} (hp : superPotential m (0,p) ≤ a) : |
| p ∈ closure {z : ℂ | (0,z) ∈ superDomain m a} := by |
| have ht : Tendsto (fun t : ℝ ↦ t • p) (𝓝[<] 1) (𝓝 p) := by |
| have hc : Continuous (fun t : ℝ ↦ t • (p : ℂ)) := continuous_id.smul continuous_const |
| have hh := hc.continuousAt (x := (1:ℝ)) |>.tendsto |
| rw [one_smul] at hh |
| exact hh.mono_left nhdsWithin_le_nhds |
| apply mem_closure_of_tendsto ht |
| filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds (show (0:ℝ)<1 by norm_num))] |
| with t ht ht0 |
| simpa only [Prod.smul_mk,smul_zero] using superPotential_radial_lt hm ha ht0.le ht hp |
| |
| lemma superSublevel_subset_bidisk {m : ℕ} (hm : 0 < m) {a : ℝ} (ha : a < 1) : |
| {z | superPotential m z ≤ a} ⊆ ball (0:ℂ) 1 ×ˢ ball (0:ℂ) 1 := by |
| intro z hz |
| exact superDomain_subset_bidisk hm (le_refl 1) (lt_of_le_of_lt hz ha) |
| |
| lemma superSublevel_widen {m : ℕ} (hm : 0 < m) {a : ℝ} {W : Set (ℂ × ℂ)} |
| (hW : IsOpen W) (hsub : {z | superPotential m z ≤ a} ⊆ W) : |
| ∃ b : ℝ, a < b ∧ superDomain m b ⊆ W := by |
| let K := {z | superPotential m z ≤ a+1} \ W |
| have hK : IsCompact K := (superSublevel_compact hm (a+1)).diff hW |
| by_cases hne : K.Nonempty |
| · obtain ⟨p,hp,hmin⟩ := hK.exists_isMinOn hne (superPotential_contDiff m (k := 0)).continuous.continuousOn |
| have hap : a < superPotential m p := by |
| by_contra h |
| exact hp.2 (hsub (le_of_not_gt h)) |
| refine ⟨min (superPotential m p) (a+1),lt_min hap (by linarith),?_⟩ |
| intro z hz |
| change superPotential m z < _ at hz |
| by_contra h |
| exact not_lt_of_ge (hmin ⟨le_of_lt (lt_of_lt_of_le hz (min_le_right _ _)),h⟩) |
| (lt_of_lt_of_le hz (min_le_left _ _)) |
| · refine ⟨a+1,by linarith,?_⟩ |
| intro z hz |
| change superPotential m z < _ at hz |
| by_contra h |
| exact hne ⟨z,hz.le,h⟩ |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology |
| namespace UniversalCoverSplitting |
| |
| lemma segment_meets_inter_balls {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {p q : E} {r s : ℝ} (hi : (ball p r ∩ ball q s).Nonempty) : |
| (segment ℝ p q ∩ (ball p r ∩ ball q s)).Nonempty := by |
| obtain ⟨x,hxp,hxq⟩ := hi |
| have hr : 0 < r := (dist_nonneg : 0 ≤ dist x p).trans_lt hxp |
| have hs : 0 < s := (dist_nonneg : 0 ≤ dist x q).trans_lt hxq |
| have hd : dist p q < r+s := lt_of_le_of_lt (dist_triangle p x q) |
| (add_lt_add (by simpa only [mem_ball,dist_comm] using hxp) hxq) |
| have hrs : 0 < r+s := add_pos hr hs |
| let t : ℝ := r/(r+s) |
| have ht : 0 < t := div_pos hr hrs |
| have ht1 : t < 1 := (div_lt_one hrs).mpr (by linarith) |
| have heq : 1-t = s/(r+s) := by dsimp [t]; field_simp; ring |
| refine ⟨AffineMap.lineMap p q t,lineMap_mem_segment ℝ p q ⟨ht.le,ht1.le⟩,?_,?_⟩ |
| · rw [mem_ball,dist_lineMap_left,Real.norm_eq_abs,abs_of_pos ht] |
| have h := mul_lt_mul_of_pos_left hd ht |
| have he : t*(r+s)=r := by dsimp [t]; field_simp |
| rwa [he] at h |
| · rw [mem_ball,dist_lineMap_right,Real.norm_eq_abs,abs_of_pos (sub_pos.mpr ht1),heq] |
| have h := mul_lt_mul_of_pos_left hd (div_pos hs hrs) |
| have he : s/(r+s)*(r+s)=s := by field_simp |
| rwa [he] at h |
| |
| lemma convex_boundary_ball_overlap {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {U : Set E} (hU : Convex ℝ U) {p q : E} (hp : p ∈ closure U) (hq : q ∈ closure U) |
| {r s : ℝ} (hi : (ball p r ∩ ball q s).Nonempty) : |
| ((ball p r ∩ ball q s) ∩ U).Nonempty := by |
| obtain ⟨x,hxs,hxi⟩ := segment_meets_inter_balls hi |
| have hxc := hU.closure.segment_subset hp hq hxs |
| obtain ⟨y,hyi,hyU⟩ := mem_closure_iff_nhds.mp hxc (ball p r ∩ ball q s) |
| ((isOpen_ball.inter isOpen_ball).mem_nhds hxi) |
| exact ⟨y,hyi,hyU⟩ |
| |
| 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} |
| |
| lemma box_extend_closure {U : Set (ℂ × ℂ)} (hU : IsOpen U) (hUc : Convex ℝ U) |
| {f : ℂ × ℂ → X.Carrier} (hf : IsBox S U f) |
| (hext : ∀ p ∈ frontier U, ∃ V : Set (ℂ × ℂ), IsOpen V ∧ p ∈ V ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S V F ∧ EqOn F f (V ∩ U)) : |
| ∃ W : Set (ℂ × ℂ), IsOpen W ∧ closure U ⊆ W ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S W F ∧ EqOn F f U := by |
| classical |
| have hb (p : frontier U) : ∃ a : ℝ, 0 < a ∧ |
| ∃ F : ℂ × ℂ → X.Carrier, IsBox S (ball (p:ℂ × ℂ) a) F ∧ |
| EqOn F f (ball (p:ℂ × ℂ) a ∩ U) := by |
| obtain ⟨V,hV,hp,F,hF,he⟩ := hext p p.property |
| obtain ⟨a,ha,haV⟩ := Metric.mem_nhds_iff.mp (hV.mem_nhds hp) |
| exact ⟨a,ha,F,hF.mono haV,he.mono (inter_subset_inter_left U haV)⟩ |
| choose a ha G hG hGf using hb |
| have hGG (p q : frontier U) : EqOn (G p) (G q) (ball (p:ℂ × ℂ) (a p) ∩ ball (q:ℂ × ℂ) (a q)) := by |
| by_cases hne : (ball (p:ℂ × ℂ) (a p) ∩ ball (q:ℂ × ℂ) (a q)).Nonempty |
| · obtain ⟨z,hzi,hzU⟩ := convex_boundary_ball_overlap hUc |
| (frontier_subset_closure p.property) (frontier_subset_closure q.property) hne |
| apply holomorphic_manifold_eqOn_of_preconnected_of_eventuallyEq |
| (isOpen_ball.inter isOpen_ball) ((convex_ball _ _).inter (convex_ball _ _)).isPreconnected |
| ((hG p).1.mono inter_subset_left) ((hG q).1.mono inter_subset_right) hzi |
| filter_upwards [(isOpen_ball.inter isOpen_ball |>.inter hU).mem_nhds ⟨hzi,hzU⟩] with w hw |
| exact (hGf p ⟨hw.1.1,hw.2⟩).trans (hGf q ⟨hw.1.2,hw.2⟩).symm |
| · intro z hz |
| exact (hne ⟨z,hz⟩).elim |
| let V : Option (frontier U) → Set (ℂ × ℂ) |
| | none => U |
| | some p => ball (p:ℂ × ℂ) (a p) |
| let G' : Option (frontier U) → ℂ × ℂ → X.Carrier |
| | none => f |
| | some p => G p |
| have hVo : ∀ i, IsOpen (V i) := by intro i; cases i with |
| | none => exact hU |
| | some p => exact isOpen_ball |
| have hGb : ∀ i, IsBox S (V i) (G' i) := by intro i; cases i with |
| | none => exact hf |
| | some p => exact hG p |
| have hcompat : ∀ i j, EqOn (G' i) (G' j) (V i ∩ V j) := by |
| intro i j |
| cases i with |
| | none => |
| cases j with |
| | none => exact fun _ _ ↦ rfl |
| | some q => exact fun z hz ↦ (hGf q ⟨hz.2,hz.1⟩).symm |
| | some p => |
| cases j with |
| | none => exact hGf p |
| | some q => exact hGG p q |
| obtain ⟨F,hF,he⟩ := glue_boxes V hVo G' hGb hcompat (f 0) |
| refine ⟨⋃ i, V i,isOpen_iUnion hVo,?_,F,hF,he none⟩ |
| intro p hp |
| by_cases hpU : p ∈ U |
| · exact mem_iUnion.mpr ⟨none,hpU⟩ |
| · have hpf : p ∈ frontier U := ⟨hp,by simpa only [hU.interior_eq] using hpU⟩ |
| exact mem_iUnion.mpr ⟨some ⟨p,hpf⟩,mem_ball_self (ha ⟨p,hpf⟩)⟩ |
|
|
| end UniversalCoverSplitting |
|
|
| end |
|
|
| end OAI |
|
|
