| import OAI.Geometry.KahlerSplitting.CompactCurves |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Manifold ContDiff |
| open Set |
| open Set Topology Filter Metric |
| open Set MeasureTheory Metric Filter Topology |
| open scoped Interval |
| open Set MeasureTheory |
| open scoped ENNReal |
| open Set Metric |
| open scoped ENNReal NNReal |
| open Set Metric Filter Topology |
| open scoped Manifold |
| open scoped Manifold ENNReal NNReal |
| open Set Metric Filter Topology MeasureTheory |
| open Complex |
| open Set Filter Topology |
|
|
| namespace UniversalCoverSplitting |
|
|
| theorem compact_family_trapping {M T : Type*} |
| [TopologicalSpace M] [TopologicalSpace T] [CompactSpace T] |
| (H : ℝ → T → M) (hH : Continuous (Function.uncurry H)) |
| {K V : Set M} (hK : IsClosed K) (hV : IsOpen V) (hVK : V ⊆ K) |
| {b : ℝ} (_hb : 0 ≤ b) (hzero : ∀ u, H 0 u ∈ K) |
| (hcontrol : ∀ t ∈ Ico 0 b, |
| (∀ s ∈ Icc 0 t, ∀ u, H s u ∈ K) → ∀ u, H t u ∈ V) : |
| ∀ t ∈ Icc 0 b, ∀ u, H t u ∈ K := by |
| let A : Set ℝ := {t | ∀ u, H t u ∈ K} |
| have hA : IsClosed A := by |
| have he : A = ⋂ u : T, (fun t ↦ H t u) ⁻¹' K := by ext t; simp [A] |
| rw [he] |
| exact isClosed_iInter (fun u ↦ hK.preimage (hH.comp (continuous_id.prodMk continuous_const))) |
| have hall : Icc 0 b ⊆ A := by |
| apply (hA.inter isClosed_Icc).Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset hzero |
| intro t ht hst |
| have hc := hcontrol t ht (fun s hs u ↦ hst hs u) |
| have he : ∀ᶠ s in 𝓝 t, ∀ u : T, H s u ∈ V := by |
| have hh := isCompact_univ.eventually_forall_of_forall_eventually |
| (x₀ := t) (K := (univ : Set T)) (P := fun s u ↦ H s u ∈ V) |
| (fun u _ ↦ hH.continuousAt (hV.mem_nhds (hc u))) |
| simpa only [mem_univ,forall_const] using hh |
| exact nhdsWithin_le_nhds (he.mono (fun s hs u ↦ hVK (hs u))) |
| exact fun t ht u ↦ hall ht u |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology Complex |
| namespace UniversalCoverSplitting |
| variable {M E : Type*} [TopologicalSpace M] [T2Space M] |
| [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] [ProperSpace E] |
|
|
| theorem punctured_chart_trapping (e : OpenPartialHomeomorph M E) |
| {f : ℂ → M} {c : E} {R r L : ℝ} |
| (_hR : 0 < R) (_hr : 0 < r) (hL : 0 ≤ L) |
| (hchart : closedBall c R ⊆ e.target) |
| (hc : ContinuousOn f {z | 0 < ‖z‖ ∧ ‖z‖ ≤ r}) |
| (houter : ∀ z ∈ sphere (0:ℂ) r, f z ∈ e.source ∧ e (f z) ∈ closedBall c R) |
| (hmargin : dist (e (f (r:ℂ))) c + 24 * L < R) |
| (hd : ∀ z, 0 < ‖z‖ → ‖z‖ ≤ r → |
| f z ∈ e.symm '' closedBall c R → DifferentiableAt ℂ (e ∘ f) z) |
| (hb : ∀ z, 0 < ‖z‖ → ‖z‖ ≤ r → |
| f z ∈ e.symm '' closedBall c R → ‖deriv (e ∘ f) z‖ * ‖z‖ ≤ L) : |
| ∀ z, 0 < ‖z‖ → ‖z‖ ≤ r → |
| f z ∈ e.source ∧ e (f z) ∈ closedBall c R := by |
| let K := e.symm '' closedBall c R |
| let V := e.source ∩ e ⁻¹' ball c R |
| have hK : IsClosed K := |
| ((isCompact_closedBall c R).image_of_continuousOn (e.continuousOn_symm.mono hchart)).isClosed |
| have hV : IsOpen V := e.isOpen_inter_preimage isOpen_ball |
| have hVK : V ⊆ K := by |
| intro x hx |
| exact ⟨e x,ball_subset_closedBall hx.2,e.left_inv hx.1⟩ |
| have hmem (x : M) (hx : x ∈ K) : x ∈ e.source ∧ e x ∈ closedBall c R := by |
| obtain ⟨w,hw,rfl⟩ := hx |
| exact ⟨e.map_target (hchart hw), by rwa [e.right_inv (hchart hw)]⟩ |
| have hring (a : ℝ) (ha : 0 < a) (har : a ≤ r) : |
| ∀ z, a ≤ ‖z‖ → ‖z‖ ≤ r → f z ∈ K := by |
| let T := sphere (0:ℂ) 1 |
| let : CompactSpace T := isCompact_iff_compactSpace.mp (isCompact_sphere (0:ℂ) 1) |
| let ρ : ℝ → ℝ := fun t ↦ max a (min (r-t) r) |
| have hρ : Continuous ρ := continuous_const.max ((continuous_const.sub continuous_id).min continuous_const) |
| have hρlo (t : ℝ) : a ≤ ρ t := le_max_left _ _ |
| have hρhi (t : ℝ) : ρ t ≤ r := max_le har (min_le_right _ _) |
| have hρeq (t : ℝ) (ht : t ∈ Icc 0 (r-a)) : ρ t = r-t := by |
| dsimp [ρ] |
| rw [min_eq_left (by linarith [ht.1]),max_eq_right (by linarith [ht.2])] |
| let H : ℝ → T → M := fun t u ↦ f ((ρ t : ℂ) * (u:ℂ)) |
| have harg (t : ℝ) (u : T) : ‖(ρ t : ℂ) * (u:ℂ)‖ = ρ t := by |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (ha.trans_le (hρlo t)), |
| norm_eq_of_mem_sphere,mul_one] |
| have hH : Continuous (Function.uncurry H) := by |
| apply hc.comp_continuous |
| · exact ((Complex.continuous_ofReal.comp (hρ.comp continuous_fst)).mul |
| (continuous_subtype_val.comp continuous_snd)) |
| · intro p |
| exact ⟨by rw [harg]; exact ha.trans_le (hρlo p.1), by rw [harg]; exact hρhi p.1⟩ |
| have hzero : ∀ u, H 0 u ∈ K := by |
| intro u |
| have hn : ‖(ρ 0 : ℂ) * (u:ℂ)‖ = r := by rw [harg,hρeq 0 ⟨le_rfl,sub_nonneg.mpr har⟩,sub_zero] |
| have ho := houter ((ρ 0:ℂ)*(u:ℂ)) (by simpa only [mem_sphere,dist_zero_right] using hn) |
| exact ⟨e _,ho.2,e.left_inv ho.1⟩ |
| have hcontrol : ∀ t ∈ Ico 0 (r-a), |
| (∀ s ∈ Icc 0 t, ∀ u, H s u ∈ K) → ∀ u, H t u ∈ V := by |
| intro t ht hprev u |
| have ht' : t ∈ Icc 0 (r-a) := ⟨ht.1,ht.2.le⟩ |
| have hrt : 0 < r-t := by linarith [ht.2] |
| have hrtR : r-t ≤ r := by linarith [ht.1] |
| have hann : ∀ z ∈ closedBall (0:ℂ) r \ ball 0 (r-t), f z ∈ K := by |
| intro w hw |
| have hwlo : r-t ≤ ‖w‖ := by simpa using hw.2 |
| have hwhi : ‖w‖ ≤ r := by simpa using hw.1 |
| have hwpos : 0 < ‖w‖ := hrt.trans_le hwlo |
| let v : T := ⟨(‖w‖⁻¹ : ℂ) * w, by |
| simp [T,inv_mul_cancel₀ hwpos.ne']⟩ |
| have hs : r-‖w‖ ∈ Icc 0 t := ⟨by linarith,by linarith⟩ |
| have hp := hprev (r-‖w‖) hs v |
| have hv : (ρ (r-‖w‖) : ℂ) * (v:ℂ) = w := by |
| rw [hρeq _ ⟨hs.1,hs.2.trans ht.2.le⟩] |
| dsimp [v] |
| simp only [sub_sub_cancel] |
| rw [← mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hwpos.ne'),one_mul] |
| simpa only [H,hv] using hp |
| have hfc : ContinuousOn (e ∘ f) (closedBall 0 r \ ball 0 (r-t)) := by |
| apply e.continuousOn.comp (hc.mono _) (fun w hw ↦ (hmem _ (hann w hw)).1) |
| intro w hw |
| have hlo : r-t ≤ ‖w‖ := by simpa using hw.2 |
| exact ⟨hrt.trans_le hlo,by simpa using hw.1⟩ |
| have ho := holomorphic_annulus_oscillation hrt hrtR hL hfc |
| (fun w hw ↦ hd w (hrt.trans_le (by simpa using hw.2)) (by simpa using hw.1) (hann w hw)) |
| (fun w hw ↦ hb w (hrt.trans_le (by simpa using hw.2)) (by simpa using hw.1) (hann w hw)) |
| ((ρ t:ℂ)*(u:ℂ)) (by simp only [mem_sphere,dist_zero_right]; rw [harg,hρeq t ht']) |
| have hm := hmem _ (hprev t ⟨ht.1,le_rfl⟩ u) |
| refine ⟨hm.1,?_⟩ |
| change dist (e (H t u)) c < R |
| have htri := dist_triangle (e (H t u)) (e (f (r:ℂ))) c |
| have hdist : dist (e (H t u)) (e (f (r:ℂ))) ≤ 24*L := by |
| simpa only [dist_eq_norm,Function.comp_apply,H] using ho |
| linarith |
| intro z hzlo hzhi |
| let u : T := ⟨(‖z‖⁻¹:ℂ)*z, by |
| have hzpos := ha.trans_le hzlo |
| simp [T,inv_mul_cancel₀ hzpos.ne']⟩ |
| have hs : r-‖z‖ ∈ Icc 0 (r-a) := ⟨by linarith,by linarith⟩ |
| have hg := compact_family_trapping H hH hK hV hVK (sub_nonneg.mpr har) hzero hcontrol _ hs u |
| have hv : (ρ (r-‖z‖):ℂ)*(u:ℂ) = z := by |
| rw [hρeq _ hs] |
| dsimp [u] |
| simp only [sub_sub_cancel] |
| rw [← mul_assoc,mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr (ha.trans_le hzlo).ne'),one_mul] |
| simpa only [H,hv] using hg |
| intro z hz hrz |
| exact hmem _ (hring ‖z‖ hz hrz z le_rfl hrz) |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ENNReal |
| namespace UniversalCoverSplitting |
| |
| theorem finite_density_small_ball {w : ℂ → ℝ≥0∞} {Ω : Set ℂ} |
| (hfin : (∫⁻ z in Ω, w z) ≠ (⊤ : ℝ≥0∞)) {ε : ℝ≥0∞} (hε : 0 < ε) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ D : Set ℂ, D ⊆ ball 0 δ → D ⊆ Ω → |
| (∫⁻ z in D, w z) < ε := by |
| obtain ⟨b,hb,hsmall⟩ := exists_pos_setLIntegral_lt_of_measure_lt hfin hε.ne' |
| have hv : Tendsto (fun t : ℝ ↦ volume (ball (0:ℂ) t)) (𝓝 0) (𝓝 0) := by |
| have hh := ENNReal.Tendsto.mul_const (ENNReal.Tendsto.pow (n := 2) |
| (ENNReal.continuous_ofReal.tendsto (0:ℝ))) (Or.inr (ENNReal.coe_ne_top : (NNReal.pi : ℝ≥0∞) ≠ (⊤ : ℝ≥0∞))) |
| simpa only [Complex.volume_ball,ENNReal.ofReal_zero,zero_pow (by norm_num : 2 ≠ 0),zero_mul] using hh |
| obtain ⟨a,ha,haV⟩ := Metric.nhds_basis_ball.mem_iff.mp (hv.eventually_lt_const hb) |
| let δ := a/2 |
| have hδ : 0 < δ := half_pos ha |
| have hm : volume (ball (0:ℂ) δ) < b := haV (by simpa [δ,Real.dist_eq,abs_of_pos ha] using (half_lt_self ha)) |
| refine ⟨δ,hδ,?_⟩ |
| intro D hD hDΩ |
| have hh := hsmall D ((Measure.restrict_le_self _).trans_lt ((measure_mono hD).trans_lt hm)) |
| rwa [Measure.restrict_restrict_of_subset hDΩ] at hh |
|
|
| variable {n : ℕ} {X : ComplexManifold n} |
|
|
| theorem finite_curveArea_scaled_speed [CompactSpace X.Carrier] (g : KahlerMetric X) |
| {f : ℂ → X.Carrier} {R : ℝ} (hR : 0 < R) |
| (hf : MDifferentiableOn 𝓘(ℂ,ℂ) (model n) f (ball 0 R \ {0})) |
| (ha : curveArea g f (ball 0 R \ {0}) ≠ (⊤ : ℝ≥0∞)) {ε : ℝ} (hε : 0 < ε) : |
| ∃ δ : ℝ, 0 < δ ∧ δ < R ∧ ∀ z : ℂ, 0 < ‖z‖ → ‖z‖ ≤ δ → |
| ‖z‖ * curveSpeed g f z < ε := by |
| obtain ⟨η,hη,hsmall⟩ := small_area_weighted_speed g (half_pos hε) |
| obtain ⟨a,ha',harea⟩ := finite_density_small_ball ha (ENNReal.ofReal_pos.mpr hη) |
| let δ := min a R / 4 |
| have hδ : 0 < δ := div_pos (lt_min ha' hR) (by norm_num) |
| have hδR : δ < R := by |
| have hd := min_le_right a R |
| dsimp [δ]; linarith [lt_min ha' hR] |
| refine ⟨δ,hδ,hδR,?_⟩ |
| intro z hz hzδ |
| have hc : closedBall z (‖z‖/2) ⊆ ball 0 R \ {0} := by |
| intro w hw |
| have hwz : dist w z ≤ ‖z‖/2 := hw |
| have htri := dist_triangle w z 0 |
| simp only [dist_zero_right] at htri |
| have htri' := dist_triangle z w 0 |
| simp only [dist_zero_right] at htri' |
| rw [dist_comm z w] at htri' |
| have hnorm : ‖w‖ < R := by |
| have hzR := hzδ.trans hδR.le |
| dsimp [δ] at hzδ |
| have hh := min_le_right a R |
| linarith |
| refine ⟨by simpa using hnorm,?_⟩ |
| intro hw0 |
| have he : w = 0 := hw0 |
| subst w |
| simp only [norm_zero,dist_zero_left] at htri' hwz |
| linarith |
| have hcsmall : ball z (‖z‖/2) ⊆ ball (0:ℂ) a := by |
| intro w hw |
| have hwz : dist w z < ‖z‖/2 := hw |
| have htri := dist_triangle w z 0 |
| simp only [dist_zero_right] at htri |
| have hh := min_le_left a R |
| dsimp [δ] at hzδ |
| have hn : ‖w‖ < a := by linarith |
| simpa using hn |
| have he : curveArea g f (ball z (‖z‖/2)) < ENNReal.ofReal η := |
| harea _ hcsmall (ball_subset_closedBall.trans hc) |
| have hh := hsmall f (ball 0 R \ {0}) (isOpen_ball.sdiff isClosed_singleton) hf z (‖z‖/2) |
| (half_pos hz) hc he z (mem_ball_self (half_pos hz)) |
| simp only [dist_self,sub_zero] at hh |
| nlinarith |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology Complex |
| namespace UniversalCoverSplitting |
| variable {M E : Type*} [TopologicalSpace M] [T2Space M] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedSpace ℂ E] |
| [IsScalarTower ℝ ℂ E] [ProperSpace E] |
| |
| theorem chart_trapping_on_circle (e : OpenPartialHomeomorph M E) |
| {f : ℂ → M} {c : E} {R r L : ℝ} |
| (hR : 0 < R) (hr : 0 < r) (hL : 0 ≤ L) |
| (he : closedBall c R ⊆ e.target) |
| (hc : ContinuousOn f (sphere (0:ℂ) r)) |
| (hzero : f (r:ℂ) ∈ e.source ∧ dist (e (f (r:ℂ))) c < R - 8*L) |
| (hd : ∀ z ∈ sphere (0:ℂ) r, f z ∈ e.symm '' closedBall c R → |
| DifferentiableAt ℂ (e ∘ f) z) |
| (hb : ∀ z ∈ sphere (0:ℂ) r, f z ∈ e.symm '' closedBall c R → |
| ‖deriv (e ∘ f) z‖ * r ≤ L) : |
| ∀ z ∈ sphere (0:ℂ) r, f z ∈ e.source ∧ e (f z) ∈ ball c R := by |
| have hγ : ContinuousOn (fun θ : ℝ ↦ f (circleMap 0 r θ)) (Icc 0 (2*Real.pi)) := |
| hc.comp (continuous_circleMap 0 r).continuousOn (fun θ _ ↦ circleMap_mem_sphere 0 hr.le θ) |
| have hinit : f (circleMap 0 r 0) ∈ e.source ∧ |
| dist (e (f (circleMap 0 r 0))) c < R - L * (2*Real.pi) := by |
| simpa [circleMap] using And.intro hzero.1 (hzero.2.trans_le (by |
| have ht : 2*Real.pi ≤ 8 := by linarith [Real.pi_lt_four] |
| nlinarith)) |
| have ht := chart_trapping_on_interval e hR hL Real.two_pi_pos.le he hγ hinit |
| (fun θ _ hk ↦ ?_) |
| · intro z hz |
| have hz' : z ∈ circleMap 0 r '' Ioc 0 (2*Real.pi) := by |
| rw [image_circleMap_Ioc,abs_of_pos hr] |
| exact hz |
| obtain ⟨θ,hθ,rfl⟩ := hz' |
| exact ht θ ⟨hθ.1.le,hθ.2⟩ |
| have hm := circleMap_mem_sphere (0:ℂ) hr.le θ |
| refine ⟨(circleMap 0 r θ * I) • deriv (e ∘ f) (circleMap 0 r θ),?_,?_⟩ |
| · exact ((hd _ hm hk).hasDerivAt.scomp θ (hasDerivAt_circleMap 0 r θ)).hasDerivWithinAt |
| · rw [norm_smul,norm_mul,norm_I,mul_one] |
| have hn : ‖circleMap 0 r θ‖ = r := by simpa using hm |
| rw [hn,mul_comm] |
| exact hb _ hm hk |
| |
| end UniversalCoverSplitting |
| |
| open Set Metric Filter Topology Complex |
| open scoped Manifold ENNReal |
| namespace UniversalCoverSplitting |
| variable {n : ℕ} {X : ComplexManifold n} |
| local instance [CompactSpace X.Carrier] : MetricSpace X.Carrier := compactManifoldMetric X |
| |
| theorem finite_curveArea_compact_chart [CompactSpace X.Carrier] (g : KahlerMetric X) |
| {f : ℂ → X.Carrier} {R : ℝ} (hR : 0 < R) |
| (hf : MDifferentiableOn 𝓘(ℂ,ℂ) (model n) f (ball 0 R \ {0})) |
| (ha : curveArea g f (ball 0 R \ {0}) ≠ (⊤ : ℝ≥0∞)) : |
| ∃ c : X.Carrier, ∃ r B : ℝ, 0 < r ∧ r < R ∧ 0 < B ∧ |
| closedBall ((extChartAt (model n) c) c) B ⊆ (extChartAt (model n) c).target ∧ |
| ∀ z : ℂ, 0 < ‖z‖ → ‖z‖ ≤ r → |
| f z ∈ (extChartAt (model n) c).source ∧ |
| (extChartAt (model n) c) (f z) ∈ closedBall ((extChartAt (model n) c) c) B := by |
| let u (j : ℕ) : ℝ := 1 / (j+1) |
| have hu (j : ℕ) : 0 < u j := by dsimp [u]; positivity |
| have hu0 : Tendsto u atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| obtain ⟨c,σ,hσ,hlim⟩ := CompactSpace.tendsto_subseq (fun j ↦ f (u j : ℂ)) |
| let e : OpenPartialHomeomorph X.Carrier (Model n) := |
| { extChartAt (model n) c with |
| continuousOn_toFun := continuousOn_extChartAt c |
| continuousOn_invFun := continuousOn_extChartAt_symm c |
| open_source := isOpen_extChartAt_source c |
| open_target := isOpen_extChartAt_target c } |
| have hc : c ∈ e.source := mem_extChartAt_source c |
| obtain ⟨B,hB,hBt⟩ := Metric.nhds_basis_closedBall.mem_iff.mp |
| (e.open_target.mem_nhds (e.map_source hc)) |
| obtain ⟨C,D,hC,_,hcomp⟩ := g.exists_chart_speed_comparison c (isCompact_closedBall (e c) B) hBt |
| let L := B/100 |
| have hL : 0 < L := div_pos hB (by norm_num) |
| obtain ⟨δ,hδ,hδR,hspeed⟩ := finite_curveArea_scaled_speed g hR hf ha (div_pos hL hC) |
| have hnear : e.source ∩ e ⁻¹' ball (e c) (B/4) ∈ 𝓝 c := |
| (e.isOpen_inter_preimage isOpen_ball).mem_nhds ⟨hc, by simpa using div_pos hB (by norm_num : (0:ℝ)<4)⟩ |
| have hnear' : ∀ᶠ j in atTop, f (u (σ j):ℂ) ∈ e.source ∩ e ⁻¹' ball (e c) (B/4) := hlim hnear |
| have he := hnear'.and ((hu0.comp hσ.tendsto_atTop).eventually_lt_const hδ) |
| obtain ⟨j,hj,hjr⟩ := he.exists |
| let r := u (σ j) |
| have hr : 0 < r := hu _ |
| have hrδ : r < δ := hjr |
| have hrR : r < R := hrδ.trans hδR |
| have hsource (z : ℂ) (hz : 0 < ‖z‖) (hzr : ‖z‖ ≤ r) : z ∈ ball 0 R \ {0} := by |
| refine ⟨by simpa using hzr.trans_lt hrR,?_⟩ |
| simpa only [mem_singleton_iff] using (norm_pos_iff.mp hz) |
| have hK (x : X.Carrier) (hx : x ∈ e.symm '' closedBall (e c) B) : |
| x ∈ e.source ∧ e x ∈ closedBall (e c) B := by |
| obtain ⟨v,hv,rfl⟩ := hx |
| exact ⟨e.map_target (hBt hv), by rwa [e.right_inv (hBt hv)]⟩ |
| have hdiff (z : ℂ) (hz : 0 < ‖z‖) (hzr : ‖z‖ ≤ r) |
| (hk : f z ∈ e.symm '' closedBall (e c) B) : DifferentiableAt ℂ (e ∘ f) z := by |
| have hfd := hf.mdifferentiableAt ((isOpen_ball.sdiff isClosed_singleton).mem_nhds (hsource z hz hzr)) |
| have hed : MDifferentiableAt (model n) (model n) e (f z) := |
| mdifferentiableAt_extChartAt (by simpa only [e,extChartAt_source] using (hK _ hk).1) |
| exact (hed.comp z hfd).differentiableAt |
| have hbound (z : ℂ) (hz : 0 < ‖z‖) (hzr : ‖z‖ ≤ r) |
| (hk : f z ∈ e.symm '' closedBall (e c) B) : ‖deriv (e ∘ f) z‖ * ‖z‖ ≤ L := by |
| have hfd := hf.mdifferentiableAt ((isOpen_ball.sdiff isClosed_singleton).mem_nhds (hsource z hz hzr)) |
| have hnorm : ‖deriv (e ∘ f) z‖ ≤ C * curveSpeed g f z := by |
| change ‖deriv ((extChartAt (model n) c) ∘ f) z‖ ≤ C * curveSpeed g f z |
| rw [chart_deriv_curve hfd c (hK _ hk).1] |
| exact (hcomp (f z) (hK _ hk).1 (hK _ hk).2 (mfderiv 𝓘(ℂ,ℂ) (model n) f z (1:ℂ))).1 |
| have hh := hspeed z hz (hzr.trans hrδ.le) |
| have hm := mul_lt_mul_of_pos_left hh hC |
| rw [mul_div_cancel₀ _ hC.ne'] at hm |
| nlinarith |
| have hcircle : ∀ z ∈ sphere (0:ℂ) r, f z ∈ e.source ∧ e (f z) ∈ ball (e c) B := by |
| apply chart_trapping_on_circle e hB hr hL.le hBt |
| · apply hf.continuousOn.mono |
| intro z hz |
| have hn : ‖z‖ = r := by simpa using hz |
| exact hsource z (hn ▸ hr) hn.le |
| · refine ⟨hj.1,?_⟩ |
| have hh : dist (e (f (r:ℂ))) (e c) < B/4 := hj.2 |
| dsimp [L]; linarith |
| · intro z hz hk |
| have hn : ‖z‖ = r := by simpa using hz |
| exact hdiff z (hn ▸ hr) hn.le hk |
| · intro z hz hk |
| have hn : ‖z‖ = r := by simpa using hz |
| simpa only [hn] using hbound z (hn ▸ hr) hn.le hk |
| have htrap := punctured_chart_trapping e hB hr hL.le hBt |
| (hf.continuousOn.mono (fun z hz ↦ hsource z hz.1 hz.2)) |
| (fun z hz ↦ ⟨(hcircle z hz).1,ball_subset_closedBall (hcircle z hz).2⟩) |
| (show dist (e (f (r:ℂ))) (e c) + 24*L < B from by |
| have hh : dist (e (f (r:ℂ))) (e c) < B/4 := hj.2 |
| dsimp [L]; linarith) hdiff hbound |
| exact ⟨c,r,B,hr,hrR,hB,hBt,htrap⟩ |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology MeasureTheory Complex |
| open scoped Manifold ENNReal |
| namespace UniversalCoverSplitting |
| variable {n : ℕ} {X : ComplexManifold n} |
|
|
| theorem compact_chart_removable {f : ℂ → X.Carrier} {R r B : ℝ} |
| (_hR : 0 < R) (hr : 0 < r) (hrR : r < R) (c : X.Carrier) |
| (hBt : closedBall ((extChartAt (model n) c) c) B ⊆ (extChartAt (model n) c).target) |
| (hf : MDifferentiableOn 𝓘(ℂ,ℂ) (model n) f (ball 0 R \ {0})) |
| (ht : ∀ z : ℂ, 0 < ‖z‖ → ‖z‖ ≤ r → |
| f z ∈ (extChartAt (model n) c).source ∧ |
| (extChartAt (model n) c) (f z) ∈ closedBall ((extChartAt (model n) c) c) B) : |
| ∃ x : X.Carrier, MDifferentiableOn 𝓘(ℂ,ℂ) (model n) (Function.update f 0 x) (ball 0 R) := by |
| classical |
| let e := extChartAt (model n) c |
| let Ω := ball (0:ℂ) r \ {0} |
| have hsub : Ω ⊆ ball 0 R \ {0} := sdiff_subset_sdiff_left (ball_subset_ball hrR.le) |
| have hmem (z : ℂ) (hz : z ∈ Ω) : f z ∈ e.source ∧ e (f z) ∈ closedBall (e c) B := by |
| exact ht z (norm_pos_iff.mpr hz.2) (le_of_lt (by simpa using hz.1)) |
| have hd : DifferentiableOn ℂ (e ∘ f) Ω := by |
| intro z hz |
| have hfd := hf.mdifferentiableAt ((isOpen_ball.sdiff isClosed_singleton).mem_nhds (hsub hz)) |
| have he : MDifferentiableAt (model n) (model n) e (f z) := |
| mdifferentiableAt_extChartAt (by simpa only [e,extChartAt_source] using (hmem z hz).1) |
| exact (he.comp z hfd).differentiableAt.differentiableWithinAt |
| have hb : BddAbove (norm ∘ (e ∘ f) '' Ω) := by |
| refine ⟨B + ‖e c‖,?_⟩ |
| rintro _ ⟨z,hz,rfl⟩ |
| have ht' := norm_le_insert (e c) (e (f z)) |
| rw [norm_sub_rev] at ht' |
| have hh : ‖e (f z) - e c‖ ≤ B := by simpa only [mem_closedBall,dist_eq_norm] using (hmem z hz).2 |
| dsimp [Function.comp_def] |
| linarith |
| let b := limUnder (𝓝[≠] (0:ℂ)) (e ∘ f) |
| let q := Function.update (e ∘ f) 0 b |
| have hq : DifferentiableOn ℂ q (ball 0 r) := |
| differentiableOn_update_limUnder_of_bddAbove (ball_mem_nhds 0 hr) hd hb |
| have hq0 := hq.differentiableAt (ball_mem_nhds 0 hr) |
| have hlim : Tendsto (e ∘ f) (𝓝[≠] (0:ℂ)) (𝓝 b) := |
| continuousAt_update_same.mp hq0.continuousAt |
| have hevent : ∀ᶠ z in 𝓝[≠] (0:ℂ), e (f z) ∈ closedBall (e c) B := by |
| filter_upwards [nhdsWithin_le_nhds (ball_mem_nhds (0:ℂ) hr),self_mem_nhdsWithin] with z hz hz0 |
| exact (hmem z ⟨hz,hz0⟩).2 |
| have hbK : b ∈ closedBall (e c) B := isClosed_closedBall.mem_of_tendsto hlim hevent |
| have hbi : MDifferentiableAt (model n) (model n) e.symm b := |
| mdifferentiableOn_extChartAt_symm.mdifferentiableAt |
| ((isOpen_extChartAt_target c).mem_nhds (hBt hbK)) |
| have hqzero : q 0 = b := Function.update_self _ _ _ |
| have hcomp : MDifferentiableAt 𝓘(ℂ,ℂ) (model n) (e.symm ∘ q) 0 := by |
| apply (hqzero ▸ hbi).comp 0 hq0.mdifferentiableAt |
| have hzero : MDifferentiableAt 𝓘(ℂ,ℂ) (model n) (Function.update f 0 (e.symm b)) 0 := by |
| apply hcomp.congr_of_eventuallyEq |
| filter_upwards [ball_mem_nhds (0:ℂ) hr] with z hz |
| by_cases hz0 : z = 0 |
| · subst z |
| simp only [Function.comp_apply,hqzero,Function.update_self] |
| · simp only [Function.update_of_ne hz0,q,Function.comp_apply] |
| exact (e.left_inv (hmem z ⟨hz,hz0⟩).1).symm |
| refine ⟨e.symm b,?_⟩ |
| intro z hz |
| by_cases hz0 : z = 0 |
| · subst z; exact hzero.mdifferentiableWithinAt |
| · have hd' := hf.mdifferentiableAt ((isOpen_ball.sdiff isClosed_singleton).mem_nhds ⟨hz,hz0⟩) |
| apply (hd'.congr_of_eventuallyEq ?_).mdifferentiableWithinAt |
| filter_upwards [isOpen_compl_singleton.mem_nhds hz0] with w hw |
| exact Function.update_of_ne hw _ _ |
|
|
| theorem finite_curveArea_removable [CompactSpace X.Carrier] (g : KahlerMetric X) |
| {f : ℂ → X.Carrier} {R : ℝ} (hR : 0 < R) |
| (hf : MDifferentiableOn 𝓘(ℂ,ℂ) (model n) f (ball 0 R \ {0})) |
| (ha : curveArea g f (ball 0 R \ {0}) ≠ (⊤ : ℝ≥0∞)) : |
| ∃ x : X.Carrier, MDifferentiableOn 𝓘(ℂ,ℂ) (model n) (Function.update f 0 x) (ball 0 R) := by |
| obtain ⟨c,r,B,hr,hrR,_,hBt,ht⟩ := finite_curveArea_compact_chart g hR hf ha |
| exact compact_chart_removable hR hr hrR c hBt hf ht |
|
|
| end UniversalCoverSplitting |
|
|
| open Set Metric Filter Topology MeasureTheory |
| open scoped Manifold ENNReal |
| namespace UniversalCoverSplitting |
| variable {n : ℕ} {X : ComplexManifold n} |
| local instance [CompactSpace X.Carrier] : MetricSpace X.Carrier := compactManifoldMetric X |
|
|
| theorem locally_uniform_curveSpeed_tendsto [CompactSpace X.Carrier] (g : KahlerMetric X) |
| {f : ℕ → ℂ → X.Carrier} {F : ℂ → X.Carrier} {Ω : Set ℂ} |
| (hΩ : IsOpen Ω) (hf : ∀ j, MDifferentiableOn 𝓘(ℂ,ℂ) (model n) (f j) Ω) |
| (hconv : TendstoLocallyUniformlyOn f F atTop Ω) {p : ℂ} (hp : p ∈ Ω) : |
| Tendsto (fun j ↦ curveSpeed g (f j) p) atTop (𝓝 (curveSpeed g F p)) := by |
| have hF := manifold_holomorphic_locally_uniform_limit hΩ hf hconv |
| have hc : ContinuousOn F Ω := hconv.continuousOn |
| (Filter.Eventually.of_forall (fun j ↦ (hf j).continuousOn)).frequently |
| let c := F p |
| let e := extChartAt (model n) c |
| have hes : IsOpen e.source := isOpen_extChartAt_source c |
| have hcs : c ∈ e.source := mem_extChartAt_source c |
| obtain ⟨δ,hδ,hδsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (hes.mem_nhds hcs) |
| have hnear : Ω ∩ F ⁻¹' ball c (δ/3) ∈ 𝓝 p := inter_mem (hΩ.mem_nhds hp) |
| ((hc.continuousAt (hΩ.mem_nhds hp)) (ball_mem_nhds c (by positivity))) |
| obtain ⟨r,hr,hrsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp hnear |
| have hKΩ : closedBall p r ⊆ Ω := fun z hz ↦ (hrsub hz).1 |
| have hFsmall (z : ℂ) (hz : z ∈ closedBall p r) : dist (F z) c < δ/3 := (hrsub hz).2 |
| have hu : TendstoUniformlyOn f F atTop (closedBall p r) := |
| (tendstoLocallyUniformlyOn_iff_forall_isCompact hΩ).mp hconv _ hKΩ (isCompact_closedBall p r) |
| have hlate : ∀ᶠ j in atTop, MapsTo (f j) (ball p r) (closedBall c δ) := by |
| filter_upwards [Metric.tendstoUniformlyOn_iff.mp hu (δ/3) (by positivity)] with j hj |
| intro z hz |
| have hzK := ball_subset_closedBall hz |
| have hdist : dist (f j z) (F z) < δ/3 := by simpa only [dist_comm] using hj z hzK |
| have ht := dist_triangle (f j z) (F z) c |
| have hh := hFsmall z hzK |
| change dist (f j z) c ≤ δ |
| linarith |
| have hFmap : MapsTo F (ball p r) (closedBall c δ) := by |
| intro z hz |
| change dist (F z) c ≤ δ |
| exact (hFsmall z (ball_subset_closedBall hz)).le.trans (by linarith) |
| have heuc : UniformContinuousOn e (closedBall c δ) := |
| (isCompact_closedBall c δ).uniformContinuousOn_of_continuous ((continuousOn_extChartAt c).mono hδsub) |
| have hl : TendstoLocallyUniformlyOn (fun j ↦ e ∘ f j) (e ∘ F) atTop (ball p r) := |
| heuc.comp_tendstoLocallyUniformlyOn (hconv.mono (ball_subset_closedBall.trans hKΩ)) hFmap hlate |
| have hhol : ∀ᶠ j in atTop, DifferentiableOn ℂ (e ∘ f j) (ball p r) := by |
| filter_upwards [hlate] with j hj |
| intro z hz |
| have hfj := (hf j).mdifferentiableAt (hΩ.mem_nhds (hKΩ (ball_subset_closedBall hz))) |
| have hej : MDifferentiableAt (model n) (model n) e (f j z) := |
| mdifferentiableAt_extChartAt (by simpa only [e,extChartAt_source] using hδsub (hj hz)) |
| exact (hej.comp z hfj).differentiableAt.differentiableWithinAt |
| have hpb : p ∈ ball p r := mem_ball_self hr |
| have ht := (hl.tendsto_at hpb).prodMk ((hl.deriv hhol isOpen_ball).tendsto_at hpb) |
| rw [← nhds_prod_eq] at ht |
| have het : e (F p) ∈ (extChartAt (model n) c).target := e.map_source hcs |
| have hq := ((g.continuous_quadraticInChart c).continuousAt |
| (((isOpen_extChartAt_target c).prod isOpen_univ).mem_nhds ⟨het,mem_univ _⟩)).tendsto.comp ht |
| have hs := Real.continuous_sqrt.continuousAt.tendsto.comp hq |
| have hfp := hF.mdifferentiableAt (hΩ.mem_nhds hp) |
| dsimp only [Function.comp_def,e] at hs |
| have heq := curveSpeed_eq_chart g hfp c hcs |
| dsimp only [Function.comp_def] at heq |
| rw [← heq] at hs |
| apply Filter.Tendsto.congr' _ hs |
| filter_upwards [hlate] with j hj |
| exact (curveSpeed_eq_chart g ((hf j).mdifferentiableAt (hΩ.mem_nhds hp)) c |
| (hδsub (hj hpb))).symm |
|
|
| theorem locally_uniform_curveArea_le [CompactSpace X.Carrier] (g : KahlerMetric X) |
| {f : ℕ → ℂ → X.Carrier} {F : ℂ → X.Carrier} {Ω : Set ℂ} |
| (hΩ : IsOpen Ω) (hf : ∀ j, MDifferentiableOn 𝓘(ℂ,ℂ) (model n) (f j) Ω) |
| (hconv : TendstoLocallyUniformlyOn f F atTop Ω) {C : ℝ≥0∞} |
| (hC : ∀ j, curveArea g (f j) Ω ≤ C) : curveArea g F Ω ≤ C := by |
| let w (j : ℕ) (z : ℂ) := ENNReal.ofReal (curveSpeed g (f j) z ^ 2) |
| have hm (j : ℕ) : AEMeasurable (w j) (volume.restrict Ω) := |
| ((ENNReal.continuous_ofReal.comp_continuousOn ((curveSpeed_continuousOn g hΩ (hf j)).pow 2))).aemeasurable hΩ.measurableSet |
| have hl (z : ℂ) (hz : z ∈ Ω) : liminf (fun j ↦ w j z) atTop = |
| ENNReal.ofReal (curveSpeed g F z ^ 2) := by |
| exact (ENNReal.continuous_ofReal.continuousAt.tendsto.comp |
| ((locally_uniform_curveSpeed_tendsto g hΩ hf hconv hz).pow 2)).liminf_eq |
| calc |
| curveArea g F Ω = ∫⁻ z in Ω, liminf (fun j ↦ w j z) atTop := by |
| apply lintegral_congr_ae |
| filter_upwards [ae_restrict_mem hΩ.measurableSet] with z hz |
| exact (hl z hz).symm |
| _ ≤ liminf (fun j ↦ curveArea g (f j) Ω) atTop := lintegral_liminf_le' hm |
| _ ≤ C := by |
| exact liminf_le_of_frequently_le (Filter.Eventually.of_forall hC).frequently |
| |
| end UniversalCoverSplitting |
| |
| end |
| |
| end OAI |
| |
