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namespace OAI
section
noncomputable section
open Set Filter Topology
namespace ProductBoxTopology
variable {P A X : Type*} [TopologicalSpace P] [TopologicalSpace A] [TopologicalSpace X]
lemma continuousOn_prod_iff {f : P × A → X} {U : Set P} {V : Set A} :
ContinuousOn f (U ×ˢ V) ↔ Continuous (fun z : U × V ↦ f (z.1,z.2)) := by
constructor
· intro h
exact h.comp_continuous (continuous_subtype_val.prodMap continuous_subtype_val)
(fun z ↦ ⟨z.1.property,z.2.property⟩)
· intro h
apply continuousOn_iff_continuous_domRestrict.mpr
exact h.comp (Homeomorph.Set.prod U V).continuous
lemma continuousOn_prod_union {f : P × A → X} {U : Set P} {V W : Set A}
(hV : IsClosed V) (hW : IsClosed W)
(hfV : ContinuousOn f (U ×ˢ V)) (hfW : ContinuousOn f (U ×ˢ W)) :
ContinuousOn f (U ×ˢ (V ∪ W)) := by
let g : U × A → X := fun z ↦ f (z.1,z.2)
have hgV : ContinuousOn g (univ ×ˢ V) := hfV.comp
(continuous_subtype_val.prodMap continuous_id).continuousOn
(fun z hz ↦ ⟨z.1.property,hz.2⟩)
have hgW : ContinuousOn g (univ ×ˢ W) := hfW.comp
(continuous_subtype_val.prodMap continuous_id).continuousOn
(fun z hz ↦ ⟨z.1.property,hz.2⟩)
have hg : ContinuousOn g (univ ×ˢ (V ∪ W)) := by
rw [prod_union]
exact hgV.union_of_isClosed hgW (isClosed_univ.prod hV) (isClosed_univ.prod hW)
apply continuousOn_prod_iff.mpr
exact hg.comp_continuous (continuous_id.prodMap continuous_subtype_val)
(fun z ↦ ⟨trivial,z.2.property⟩)
def LocalFamilyContinuous (R : P → A → X) (p : P) (K : Set A) : Prop :=
∃ U : Set P, IsOpen U ∧ p ∈ U ∧ ContinuousOn (Function.uncurry R) (U ×ˢ K)
lemma LocalFamilyContinuous.mono {R : P → A → X} {p : P} {K L : Set A}
(h : LocalFamilyContinuous R p K) (hLK : L ⊆ K) : LocalFamilyContinuous R p L := by
obtain ⟨U,hU,hp,hR⟩ := h
exact ⟨U,hU,hp,hR.mono (prod_mono_right hLK)⟩
lemma LocalFamilyContinuous.union {R : P → A → X} {p : P} {K L : Set A}
(hK : IsClosed K) (hL : IsClosed L)
(hR : LocalFamilyContinuous R p K) (hS : LocalFamilyContinuous R p L) :
LocalFamilyContinuous R p (K ∪ L) := by
obtain ⟨U,hU,hp,hRu⟩ := hR
obtain ⟨V,hV,hpv,hRv⟩ := hS
exact ⟨U ∩ V,hU.inter hV,⟨hp,hpv⟩,continuousOn_prod_union hK hL
(hRu.mono (prod_mono_left inter_subset_left))
(hRv.mono (prod_mono_left inter_subset_right))⟩
lemma LocalFamilyContinuous.of_continuous {R : P → A → X} {p : P} {K : Set A}
(h : Continuous (Function.uncurry R)) : LocalFamilyContinuous R p K :=
⟨univ,isOpen_univ,trivial,h.continuousOn⟩
lemma LocalFamilyContinuous.subtype_parameter {R : P → A → X} {p : P} {K : Set A}
{U : Set P} (hU : IsOpen U) (hp : p ∈ U)
(h : LocalFamilyContinuous (fun q : U ↦ R q) ⟨p,hp⟩ K) :
LocalFamilyContinuous R p K := by
obtain ⟨W,hW,hpW,hRW⟩ := h
obtain ⟨V,hV,hVe⟩ := isOpen_induced_iff.mp hW
have hpV : p ∈ V := by change (⟨p,hp⟩ : U) ∈ Subtype.val ⁻¹' V; rwa [hVe]
refine ⟨U ∩ V,hU.inter hV,⟨hp,hpV⟩,continuousOn_prod_iff.mpr ?_⟩
have hc := continuousOn_prod_iff.mp hRW
let k : ↥(U ∩ V) → W := fun q ↦ ⟨⟨q.val,q.property.1⟩,by rw [← hVe]; exact q.property.2⟩
have hk : Continuous k := (continuous_subtype_val.subtype_mk _).subtype_mk _
exact hc.comp (hk.prodMap continuous_id)
lemma LocalFamilyContinuous.precomp {B : Type*} [TopologicalSpace B]
{R : P → A → X} {p : P} {K : Set A} {L : Set B}
(h : LocalFamilyContinuous R p K) {f : B → A} (hf : Continuous f)
(hmap : MapsTo f L K) : LocalFamilyContinuous (fun p b ↦ R p (f b)) p L := by
obtain ⟨U,hU,hp,hR⟩ := h
exact ⟨U,hU,hp,hR.comp (continuous_id.prodMap hf).continuousOn
(fun z hz ↦ ⟨hz.1,hmap hz.2⟩)⟩
lemma LocalFamilyContinuous.of_restrict {R : P → A → X} {p : P} {K : Set A}
(h : LocalFamilyContinuous (fun q (a : K) ↦ R q a) p univ) :
LocalFamilyContinuous R p K := by
obtain ⟨U,hU,hp,hR⟩ := h
refine ⟨U,hU,hp,continuousOn_prod_iff.mpr ?_⟩
exact hR.comp_continuous (continuous_subtype_val.prodMap continuous_id)
(fun z ↦ ⟨z.1.property,trivial⟩)
lemma LocalFamilyContinuous.continuousAt {R : P → A → X} {p : P}
(h : LocalFamilyContinuous R p univ) (a : A) :
ContinuousAt (Function.uncurry R) (p,a) := by
obtain ⟨U,hU,hp,hR⟩ := h
exact hR.continuousAt ((hU.prod isOpen_univ).mem_nhds ⟨hp,trivial⟩)
end ProductBoxTopology
end
end
section
noncomputable section
open Set Filter Topology
namespace ProductBoxTopology
variable {P A B X C D : Type*} [TopologicalSpace P] [TopologicalSpace A]
[TopologicalSpace B] [TopologicalSpace X] [TopologicalSpace C] [TopologicalSpace D]
lemma ProductChart.parameter_tile_local [T2Space X] [CompactSpace A] [CompactSpace B]
[PathConnectedSpace A] [PathConnectedSpace B]
{atlas : Set (OpenPartialHomeomorph X (C × D))}
(hcover : ∀ x : X, ∃ e ∈ atlas, x ∈ e.source) (e : ProductChart atlas)
{R : P → A × B → X} (hR : ∀ p, IsBox atlas (R p)) (p₀ : P) (a₀ : A) (b₀ : B)
(hbottom : LocalFamilyContinuous (fun p a ↦ R p (a,b₀)) p₀ univ)
(hleft : LocalFamilyContinuous (fun p b ↦ R p (a₀,b)) p₀ univ)
(he : ∀ w, R p₀ w ∈ e.chart.source) : LocalFamilyContinuous R p₀ univ := by
obtain ⟨U,hU,hpU,hbU⟩ := hbottom
obtain ⟨V,hV,hpV,hlV⟩ := hleft
let W := U ∩ V
have hW : IsOpen W := hU.inter hV
have hpW : p₀ ∈ W := ⟨hpU,hpV⟩
have hb : Continuous (fun z : W × A ↦ R z.1 (z.2,b₀)) :=
hbU.comp_continuous (continuous_subtype_val.prodMap continuous_id)
(fun z ↦ ⟨z.1.property.1,trivial⟩)
have hl : Continuous (fun z : W × B ↦ R z.1 (a₀,z.2)) :=
hlV.comp_continuous (continuous_subtype_val.prodMap continuous_id)
(fun z ↦ ⟨z.1.property.2,trivial⟩)
obtain ⟨T,hT,hpT,hRT⟩ := e.parameter_tile hcover (R := fun p : W ↦ R p)
(fun p ↦ hR p) ⟨p₀,hpW⟩ a₀ b₀ hb hl he
apply LocalFamilyContinuous.subtype_parameter hW hpW
refine ⟨T,hT,hpT,continuousOn_prod_iff.mpr ?_⟩
exact hRT.comp (continuous_id.prodMap continuous_subtype_val)
lemma LocalFamilyContinuous.prod_of_restrict {R : P → A × B → X} {p : P}
{U : Set A} {V : Set B}
(h : LocalFamilyContinuous (fun q (w : U × V) ↦ R q (w.1,w.2)) p univ) :
LocalFamilyContinuous R p (U ×ˢ V) := by
apply LocalFamilyContinuous.of_restrict
exact h.precomp (Homeomorph.Set.prod U V).continuous (mapsTo_univ _ _)
end ProductBoxTopology
end
end
section
noncomputable section
open Set Filter Topology
open scoped unitInterval
namespace ProductBoxTopology
variable {P X C D : Type*} [TopologicalSpace P] [TopologicalSpace X]
[TopologicalSpace C] [TopologicalSpace D]
theorem continuous_family [T2Space X]
{atlas : Set (OpenPartialHomeomorph X (C × D))}
(hcharts : ∀ x : X, ∃ e : ProductChart atlas, x ∈ e.chart.source)
{R : P → I × I → X} (hR : ∀ p, IsBox atlas (R p))
(hbottom : Continuous (fun z : P × I ↦ R z.1 (z.2,0)))
(hleft : Continuous (fun z : P × I ↦ R z.1 (0,z.2))) :
Continuous (Function.uncurry R) := by
classical
have hcover : ∀ x : X, ∃ e ∈ atlas, x ∈ e.source := by
intro x
obtain ⟨e,he⟩ := hcharts x
exact ⟨e.chart,e.mem_atlas,he⟩
have hbLocal (p : P) : LocalFamilyContinuous R p (univ ×ˢ {0}) := by
refine ⟨univ,isOpen_univ,trivial,?_⟩
have hc : Continuous (fun z : P × (I × I) ↦ R z.1 (z.2.1,0)) :=
hbottom.comp (continuous_fst.prodMk continuous_snd.fst)
apply hc.continuousOn.congr
rintro ⟨q,a,b⟩ h
have hb : b=0 := h.2.2
subst b
rfl
have hlLocal (p : P) : LocalFamilyContinuous R p ({0} ×ˢ univ) := by
refine ⟨univ,isOpen_univ,trivial,?_⟩
have hc : Continuous (fun z : P × (I × I) ↦ R z.1 (0,z.2.2)) :=
hleft.comp (continuous_fst.prodMk continuous_snd.snd)
apply hc.continuousOn.congr
rintro ⟨q,a,b⟩ h
have ha : a=0 := h.2.1
subst a
rfl
have hlocal (p₀ : P) : LocalFamilyContinuous R p₀ univ := by
let c : ProductChart atlas → Set (I × I) := fun e ↦ R p₀ ⁻¹' e.chart.source
have hc : ∀ e, IsOpen (c e) := fun e ↦ e.chart.open_source.preimage (hR p₀).1
have hcU : univ ⊆ ⋃ e, c e := by
intro w _
obtain ⟨e,he⟩ := hcharts (R p₀ w)
exact mem_iUnion.mpr ⟨e,he⟩
obtain ⟨t,ht0,htmono,⟨N,hN⟩,ht⟩ :=
exists_monotone_Icc_subset_open_cover_unitInterval_prod_self hc hcU
have hprefix : ∀ i j, LocalFamilyContinuous R p₀ (Icc 0 (t i) ×ˢ Icc 0 (t j)) := by
intro i
induction i with
| zero =>
intro j
apply (hlLocal p₀).mono
intro w hw
exact ⟨le_antisymm (ht0 ▸ hw.1.2) hw.1.1,trivial⟩
| succ i hi =>
intro j
induction j with
| zero =>
apply (hbLocal p₀).mono
intro w hw
exact ⟨trivial,le_antisymm (ht0 ▸ hw.2.2) hw.2.1⟩
| succ j hj =>
have hti : t i ≤ t (i+1) := htmono (Nat.le_succ i)
have htj : t j ≤ t (j+1) := htmono (Nat.le_succ j)
let A := Icc (t i) (t (i+1))
let B := Icc (t j) (t (j+1))
let : PathConnectedSpace A := pathConnected_Icc_unit hti
let : PathConnectedSpace B := pathConnected_Icc_unit htj
let a₀ : A := ⟨t i,le_rfl,hti⟩
let b₀ : B := ⟨t j,le_rfl,htj⟩
obtain ⟨e,he⟩ := ht i j
have htile : LocalFamilyContinuous R p₀ (A ×ˢ B) := by
apply LocalFamilyContinuous.prod_of_restrict
apply e.parameter_tile_local hcover
(fun p ↦ (hR p).precomp continuous_subtype_val continuous_subtype_val) p₀ a₀ b₀
· exact hj.precomp (continuous_subtype_val.prodMk continuous_const)
(fun a _ ↦ ⟨⟨bot_le,a.property.2⟩,⟨bot_le,le_rfl⟩⟩)
· exact (hi (j+1)).precomp (continuous_const.prodMk continuous_subtype_val)
(fun b _ ↦ ⟨⟨bot_le,le_rfl⟩,⟨bot_le,b.property.2⟩⟩)
· intro w
exact he ⟨w.1.property,w.2.property⟩
have hcl : IsClosed (Icc (0:I) (t i) ×ˢ Icc (0:I) (t (j+1))) := isClosed_Icc.prod isClosed_Icc
have hcb : IsClosed (Icc (0:I) (t (i+1)) ×ˢ Icc (0:I) (t j)) := isClosed_Icc.prod isClosed_Icc
have hct : IsClosed (A ×ˢ B) := isClosed_Icc.prod isClosed_Icc
apply (((hi (j+1)).union hcl hcb hj).union (hcl.union hcb) hct htile).mono
intro w hw
by_cases hx : w.1 ≤ t i
· exact Or.inl (Or.inl ⟨⟨hw.1.1,hx⟩,hw.2⟩)
by_cases hy : w.2 ≤ t j
· exact Or.inl (Or.inr ⟨hw.1,⟨hw.2.1,hy⟩⟩)
exact Or.inr ⟨⟨(not_le.mp hx).le,hw.1.2⟩,⟨(not_le.mp hy).le,hw.2.2⟩⟩
apply (hprefix N N).mono
intro w _
rw [hN N le_rfl]
exact ⟨⟨bot_le,le_top⟩,⟨bot_le,le_top⟩⟩
exact continuous_iff_continuousAt.mpr fun z ↦ (hlocal z.1).continuousAt z.2
end ProductBoxTopology
end
end
section
noncomputable section
open Set Metric Filter Topology
open scoped Manifold unitInterval
namespace UniversalCoverSplitting
variable {n r s : ℕ} {X : ComplexManifold n} {S : HolomorphicSplitting X r s}
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
[NormedAddCommGroup F] [NormedSpace ℂ F]
theorem ball_axes_zero [CompactSpace X.Carrier] (g : KahlerMetric X)
(h₁ : Integrable S.projection) (h₂ : Integrable (complementaryProjection S))
{α : E → X.Carrier} {β : F → X.Carrier}
(hα : IsHorizontalMap S (ball (0:E) 1) α) (hβ : IsVerticalMap S (ball (0:F) 1) β)
(hbase : α 0=β 0) :
∃ H : E × F → X.Carrier, IsBlockBox S (ball 0 1 ×ˢ ball 0 1) H ∧
(∀ p ∈ ball 0 1, H (p,0)=α p) ∧ (∀ q ∈ ball 0 1, H (0,q)=β q) := by
classical
let U : Set (E × F) := ball 0 1 ×ˢ ball 0 1
have hU : IsOpen U := isOpen_ball.prod isOpen_ball
have h0E : (0:E) ∈ ball (0:E) 1 := by simp
have h0F : (0:F) ∈ ball (0:F) 1 := by simp
have hex (p : U) := exists_radial_plaque_box g h₁ h₂ hα hβ hbase p.property.1 p.property.2
let R : (E × F) → I × I → X.Carrier := fun p ↦
if hp : p ∈ U then Classical.choose (hex ⟨p,hp⟩) else fun _ ↦ α 0
have hR (p : E × F) (hp : p ∈ U) := Classical.choose_spec (hex ⟨p,hp⟩)
have hbox (p : E × F) (hp : p ∈ U) :
ProductBoxTopology.IsBox (splittingAtlas S) (R p) := by
simpa only [R,dite_eq_left hp] using (hR p hp).1
have hbottom (p : E × F) (hp : p ∈ U) (t : I) : R p (t,0)=α (unitComplex t • p.1) := by
simpa only [R,dite_eq_left hp] using (hR p hp).2.1 t
have hleft (p : E × F) (hp : p ∈ U) (t : I) : R p (0,t)=β (unitComplex t • p.2) := by
simpa only [R,dite_eq_left hp] using (hR p hp).2.2 t
have hsm (t : I) : ‖unitComplex t‖ ≤ 1 := by rw [norm_unitComplex]; exact t.property.2
have hcB : Continuous (fun z : U × I ↦ R z.1 (z.2,0)) := by
have hmap : ∀ z : U × I, unitComplex z.2 • z.1.val.1 ∈ ball (0:E) 1 :=
fun z ↦ smul_unit_ball (hsm z.2) z.1.property.1
have hc : Continuous (fun z : U × I ↦ unitComplex z.2 • z.1.val.1) :=
(continuous_unitComplex.comp continuous_snd).smul (continuous_subtype_val.comp continuous_fst).fst
exact hα.1.continuousOn.comp_continuous hc hmap |>.congr fun z ↦ (hbottom _ z.1.property z.2).symm
have hcL : Continuous (fun z : U × I ↦ R z.1 (0,z.2)) := by
have hmap : ∀ z : U × I, unitComplex z.2 • z.1.val.2 ∈ ball (0:F) 1 :=
fun z ↦ smul_unit_ball (hsm z.2) z.1.property.2
have hc : Continuous (fun z : U × I ↦ unitComplex z.2 • z.1.val.2) :=
(continuous_unitComplex.comp continuous_snd).smul (continuous_subtype_val.comp continuous_fst).snd
exact hβ.1.continuousOn.comp_continuous hc hmap |>.congr fun z ↦ (hleft _ z.1.property z.2).symm
have hcR := ProductBoxTopology.continuous_family (splittingProductCharts_cover h₁ h₂)
(fun p : U ↦ hbox p p.property) hcB hcL
have hc : ContinuousOn (Function.uncurry R) (U ×ˢ univ) := by
apply ProductBoxTopology.continuousOn_prod_iff.mpr
exact hcR.comp (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))
have hb (t : I) : IsBlockBox S U (fun p ↦ R p (t,0)) := by
have ht := hα.comp isOpen_ball isOpen_ball
(show DifferentiableOn ℂ (fun p : E ↦ unitComplex t • p) (ball 0 1) from by fun_prop)
(fun p hp ↦ smul_unit_ball (hsm t) hp)
exact (ht.block (V := ball (0:F) 1) isOpen_ball isOpen_ball).congr hU
(fun p hp ↦ hbottom p hp t)
have hl (t : I) : IsBlockBox S U (fun p ↦ R p (0,t)) := by
have ht := hβ.comp isOpen_ball isOpen_ball
(show DifferentiableOn ℂ (fun q : F ↦ unitComplex t • q) (ball 0 1) from by fun_prop)
(fun q hq ↦ smul_unit_ball (hsm t) hq)
exact (ht.block (U := ball (0:E) 1) isOpen_ball isOpen_ball).congr hU
(fun p hp ↦ hleft p hp t)
have hcover : ∀ x : X.Carrier, ∃ e ∈ splittingAtlas S, x ∈ e.source := by
intro x
obtain ⟨e,he,_⟩ := exists_splittingChart S h₁ h₂ x
exact ⟨e.chart,⟨e,rfl⟩,he⟩
refine ⟨fun p ↦ R p (1,1),block_endpoint_of_family h₁ h₂ hU hbox hc hb hl,?_,?_⟩
· intro p hp
have hpu : (p,0) ∈ U := ⟨hp,h0F⟩
have he := ProductBoxTopology.unique (splittingAtlas S) hcover (hbox (p,0) hpu)
((hbox (p,0) hpu).precomp continuous_id (continuous_const : Continuous (fun _ : I ↦ (0:I))))
(fun _ ↦ rfl) (fun t ↦ by
change R (p,0) (0,t)=R (p,0) (0,0)
rw [hleft _ hpu,hleft _ hpu]; simp)
have h := congrFun he (1,1)
change R (p,0) (1,1)=R (p,0) (1,0) at h
rw [hbottom _ hpu] at h
simpa [unitComplex] using h
· intro q hq
have hqu : (0,q) ∈ U := ⟨h0E,hq⟩
have he := ProductBoxTopology.unique (splittingAtlas S) hcover (hbox (0,q) hqu)
((hbox (0,q) hqu).precomp (continuous_const : Continuous (fun _ : I ↦ (0:I))) continuous_id)
(fun t ↦ by
change R (0,q) (t,0)=R (0,q) (0,0)
rw [hbottom _ hqu,hbottom _ hqu]; simp) (fun _ ↦ rfl)
have h := congrFun he (1,1)
change R (0,q) (1,1)=R (0,q) (0,1) at h
rw [hleft _ hqu] at h
simpa [unitComplex] using h
end UniversalCoverSplitting
end
end
end OAI
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