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openAI-math-dataset / lean /OAI /Geometry /Cannon /Conformal /Normalizer.lean
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Add lean source and binary files (part 66)
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import Mathlib
import OAI.Geometry.Cannon.Conformal.LinearZ
import OAI.Geometry.Cannon.Conformal.BoundedCrossRatios
import OAI.Geometry.Cannon.Conformal.PlaneWeakQSBounded
import OAI.Geometry.Cannon.Actions.Height
namespace OAI
section
section
section
section
section
noncomputable section
noncomputable section
open Set Filter Topology
open scoped OnePoint
noncomputable section
open Set Filter Topology
namespace CannonMobius
open Set Filter Topology
open scoped OnePoint
lemma biholomorphic_stabilizer {f : Sphere ≃ₜ Sphere} (hf : f ∈ biholomorphic)
(ht : tripleMap f standard=standard) : f=1 := by
have hi (i : Fin 3) : f (std i)=std i := congrFun (congrArg Subtype.val ht) i
obtain ⟨a,b,ha,he⟩ := analytic_fix_infty_eq_affine f (hi 0) hf
have h0 := hi 1
have h1 := hi 2
rw [he] at h0 h1
change affine a b ha (0:ℂ)=((0:ℂ):Sphere) at h0
change affine a b ha (1:ℂ)=((1:ℂ):Sphere) at h1
rw [affine_coe] at h0 h1
have hb : b=0 := by simpa using h0
have ha1 : a=1 := by simpa [hb] using h1
rw [he]
apply Homeomorph.ext
intro p
induction p using OnePoint.rec with
| infty => rfl
| coe z => simp [affine,affinePlane,ha1,hb]
lemma triple_normalizer_unique {f g : Sphere ≃ₜ Sphere} (hf : f ∈ biholomorphic)
(hg : g ∈ biholomorphic) (he : tripleMap f standard=tripleMap g standard) : f=g := by
have h : tripleMap (g⁻¹*f) standard=standard := by
rw [tripleMap_mul,he]
apply Subtype.ext
funext i
exact g.symm_apply_apply _
have hh := biholomorphic_stabilizer (biholomorphic.mul_mem (biholomorphic.inv_mem hg) hf) h
exact (inv_mul_eq_one.mp hh).symm
end CannonMobius
namespace CannonMobius
open Set Filter Topology CannonH3
open scoped OnePoint
lemma reflection_inv : boundaryReflection⁻¹=boundaryReflection :=
inv_eq_of_mul_eq_one_left boundaryReflection_sq
lemma reflection_affine (a b : ℂ) (ha : a ≠ 0) :
boundaryReflection*affine a b ha*boundaryReflection=affine (star a) (star b) (star_ne_zero.mpr ha) := by
apply Homeomorph.ext
intro p
induction p using OnePoint.rec with
| infty => rfl
| coe z =>
change ((star (a*star z+b):ℂ):Sphere)=((star a*z+star b:ℂ):Sphere)
rw [star_add,star_mul,star_star,mul_comm z]
lemma reflection_inversion : boundaryReflection*inversion*boundaryReflection=inversion := by
apply Homeomorph.ext
intro p
induction p using OnePoint.rec with
| infty =>
change boundaryReflection (inv ∞)=inv ∞
rw [inv_infty]
change ((star (0:ℂ):ℂ):Sphere)=((0:ℂ):Sphere)
simp
| coe z =>
by_cases hz : z=0
· subst z
change boundaryReflection (inv ((star (0:ℂ):ℂ):Sphere))=inv ((0:ℂ):Sphere)
simp only [star_zero,inv_zero]
rfl
· change boundaryReflection (inv ((star z:ℂ):Sphere))=inv (z:Sphere)
rw [inv_coe (star_ne_zero.mpr hz),inv_coe hz]
change ((star ((star z)⁻¹):ℂ):Sphere)=((z⁻¹:ℂ):Sphere)
rw [star_inv₀,star_star]
lemma reflection_conjugate_biholomorphic {f : Sphere ≃ₜ Sphere} (hf : f ∈ biholomorphic) :
boundaryReflection*f*boundaryReflection ∈ biholomorphic := by
let C : MulAut (Sphere ≃ₜ Sphere) := MulAut.conj boundaryReflection
have he (g : Sphere ≃ₜ Sphere) : C g=boundaryReflection*g*boundaryReflection := by
simp [C,MulAut.conj_apply,reflection_inv]
have ha (a b : ℂ) (h : a ≠ 0) : C (affine a b h) ∈ biholomorphic := by
rw [he,reflection_affine]
exact affine_biholomorphic _ _ _
have hi : C inversion ∈ biholomorphic := by rw [he,reflection_inversion]; exact inversion_biholomorphic
rw [← he]
rcases biholomorphic_normal_form f hf with ⟨a,b,h,rfl⟩|⟨p,a,b,h,rfl⟩
· exact ha a b h
· rw [map_mul,map_mul]
exact biholomorphic.mul_mem (biholomorphic.mul_mem (ha 1 p one_ne_zero) hi) (ha a b h)
lemma reflection_standard : tripleMap boundaryReflection standard=standard := by
apply Subtype.ext
funext i
fin_cases i <;> simp [tripleMap,standard,std,boundaryReflection]
def normalizer (t : Triple) : biholomorphic :=
⟨(exists_triple_normalizer t).choose,(exists_triple_normalizer t).choose_spec.1⟩
lemma normalizer_spec (t : Triple) : tripleMap (normalizer t).val standard=t :=
(exists_triple_normalizer t).choose_spec.2
def center (t : Triple) : Half :=
interior ⟨(normalizer t).val,biholomorphic_mem_boundaryGroup (normalizer t).property⟩ basepoint
lemma center_orbit (f : Sphere ≃ₜ Sphere) (hf : f ∈ biholomorphic) :
center (tripleMap f standard)=interior ⟨f,biholomorphic_mem_boundaryGroup hf⟩ basepoint := by
have he : (normalizer (tripleMap f standard)).val=f :=
triple_normalizer_unique (normalizer _).property hf (normalizer_spec _)
unfold center
congr 2
exact Subtype.ext he
lemma center_biholomorphic (f : Sphere ≃ₜ Sphere) (hf : f ∈ biholomorphic) (t : Triple) :
center (tripleMap f t)=interior ⟨f,biholomorphic_mem_boundaryGroup hf⟩ (center t) := by
have he : tripleMap (f*(normalizer t).val) standard=tripleMap f t := by
rw [tripleMap_mul,normalizer_spec]
rw [← he,center_orbit _ (biholomorphic.mul_mem hf (normalizer t).property)]
change CannonH3.interior (⟨f,biholomorphic_mem_boundaryGroup hf⟩*
⟨(normalizer t).val,biholomorphic_mem_boundaryGroup (normalizer t).property⟩) basepoint=_
rw [interior_mul]
rfl
lemma center_reflection (t : Triple) : center (tripleMap boundaryReflection t)=reflection (center t) := by
let f := (normalizer t).val
have hf : f ∈ biholomorphic := (normalizer t).property
have hfc := reflection_conjugate_biholomorphic hf
have he : tripleMap (boundaryReflection*f*boundaryReflection) standard=tripleMap boundaryReflection t := by
rw [tripleMap_mul,tripleMap_mul,reflection_standard,normalizer_spec]
rw [← he,center_orbit _ hfc]
change CannonH3.interior (⟨boundaryReflection,reflection_mem_boundaryGroup⟩*
⟨f,biholomorphic_mem_boundaryGroup hf⟩*⟨boundaryReflection,reflection_mem_boundaryGroup⟩) basepoint=_
rw [interior_mul,interior_mul,interior_reflection]
have hb : reflection basepoint=basepoint := by
apply Prod.ext
· change star (0:ℂ)=0
exact star_zero _
· rfl
change reflection (CannonH3.interior ⟨f,biholomorphic_mem_boundaryGroup hf⟩ (reflection basepoint))=_
rw [hb]
rfl
lemma center_conformal (f : Sphere ≃ₜ Sphere)
(hf : f ∈ biholomorphic ∨ boundaryReflection*f ∈ biholomorphic) (t : Triple) :
center (tripleMap f t)=interior ⟨f,conformal_or_anticonformal_mem hf⟩ (center t) := by
rcases hf with hf|hf
· exact center_biholomorphic f hf t
· have he : boundaryReflection*(boundaryReflection*f)=f := by rw [← mul_assoc,boundaryReflection_sq,one_mul]
have hs : (⟨f,conformal_or_anticonformal_mem (Or.inr hf)⟩ : boundaryGroup)=
⟨boundaryReflection,reflection_mem_boundaryGroup⟩*
⟨boundaryReflection*f,biholomorphic_mem_boundaryGroup hf⟩ := Subtype.ext he.symm
rw [hs,interior_mul,interior_reflection]
conv_lhs => rw [← he,tripleMap_mul,center_reflection,center_biholomorphic _ hf]
rfl
lemma center_surjective : Function.Surjective center := by
intro x
have ht : (x.2.val:ℂ) ≠ 0 := by exact_mod_cast x.2.property.ne'
refine ⟨tripleMap (affine (x.2.val:ℂ) x.1 ht) standard,?_⟩
rw [center_orbit _ (affine_biholomorphic _ _ _),interior_affine]
apply Prod.ext
· change (x.2.val:ℂ)*0+x.1=x.1
simp
· apply Subtype.ext
change ‖(x.2.val:ℂ)‖*1=x.2.val
simp [Complex.norm_real,abs_of_pos x.2.property]
end CannonMobius
end
end
end
end
noncomputable section
namespace CannonMobius
open Set Filter Topology MeasureTheory CannonMetricQC CannonConformal CannonQM CannonH3
open scoped OnePoint ComplexConjugate
lemma sendToInfty_biholomorphic (p : Sphere) : sendToInfty p ∈ biholomorphic := by
by_cases hp : p=∞
· simp [sendToInfty,hp]
· rw [sendToInfty,ite_eq_right hp]
exact biholomorphic.mul_mem inversion_biholomorphic (affine_biholomorphic 1 (-coordinate p) one_ne_zero)
lemma ae_conformal_post_biholomorphic (f M : Sphere ≃ₜ Sphere) (hM : M ∈ biholomorphic)
(hf : ∀ᵐ z : ℂ, DifferentiableAt ℝ (inChart false false f) z ∧
IsConformalMap (fderiv ℝ (inChart false false f) z)) :
∀ᵐ z : ℂ, DifferentiableAt ℝ (inChart false false (M*f)) z ∧
IsConformalMap (fderiv ℝ (inChart false false (M*f)) z) := by
filter_upwards [hf,chartDomain_ae f false false,chartDomain_ae (M*f) false false]
with z hz hzdom hMz
have hMd : inChart false false f z ∈ chartDomain false false M := by
change M (coordinate (f (z:Sphere)):Sphere) ≠ ∞
have hzfinite : f (z:Sphere) ≠ ∞ := hzdom
rw [coe_coordinate hzfinite]
exact hMz
have hdM := (analytic_differentiableAt hM.1 false false hMd).restrictScalars ℝ
have hm := analytic_chart_conformal M hM hMd
have hc := (hdM.hasFDerivAt.comp z hz.1.hasFDerivAt).congr_of_eventuallyEq
(chart_mul_eventually M f hzdom)
exact ⟨hc.differentiableAt,hc.fderiv ▸ hm.comp hz.2⟩
lemma restriction_affine_or_anti (f : Sphere ≃ₜ Sphere) (hi : f ∞=∞)
(h : (∃ a b : ℂ, a ≠ 0 ∧ ∀ z, finiteRestriction f hi z=a*z+b) ∨
(∃ a b : ℂ, a ≠ 0 ∧ ∀ z, finiteRestriction f hi z=a*conj z+b)) :
f ∈ biholomorphic ∨ boundaryReflection*f ∈ biholomorphic := by
rcases h with ⟨a,b,ha,he⟩ | ⟨a,b,ha,he⟩
· left
have hEq : f=affine a b ha := by
apply Homeomorph.ext
intro p
induction p using OnePoint.rec with
| infty => exact hi
| coe z =>
have hz := congrArg (fun w : ℂ => (w:Sphere)) (he z)
exact (coe_coordinate (coe_map_ne_infty f hi z)).symm.trans hz
rw [hEq]
exact affine_biholomorphic a b ha
· right
have hEq : f=affine a b ha*boundaryReflection := by
apply Homeomorph.ext
intro p
induction p using OnePoint.rec with
| infty => exact hi
| coe z =>
have hz := congrArg (fun w : ℂ => (w:Sphere)) (he z)
exact (coe_coordinate (coe_map_ne_infty f hi z)).symm.trans hz
rw [hEq,← mul_assoc,reflection_affine]
exact affine_biholomorphic _ _ _
theorem sphere_ae_conformal_rigidity (f : Sphere ≃ₜ Sphere)
(hf : BoundedCrossRatios f) (hi : BoundedCrossRatios f.symm)
(hd : ∀ᵐ z : ℂ, DifferentiableAt ℝ (inChart false false f) z ∧
IsConformalMap (fderiv ℝ (inChart false false f) z)) :
f ∈ biholomorphic ∨ boundaryReflection*f ∈ biholomorphic := by
let M := sendToInfty (f ∞)
have hM : M ∈ biholomorphic := sendToInfty_biholomorphic _
have hMc : Controlled M (OrderIso.refl _) := sendToInfty_controlled _
have hInf : (M*f) ∞=∞ := sendToInfty_apply _
have hC : BoundedCrossRatios (M*f) := hMc.boundedCrossRatios.comp hf f.injective
have hCi : BoundedCrossRatios (M*f).symm :=
hi.comp (hMc.inverse M).boundedCrossRatios M.symm.injective
let P := finiteRestriction (M*f) hInf
obtain ⟨H,hH,hP⟩ := BoundedCrossRatios.plane_weakQS P
(boundedCrossRatios_restriction (M*f) hInf hC)
obtain ⟨J,hJ,hPi⟩ := BoundedCrossRatios.plane_weakQS P.symm
(boundedCrossRatios_restriction (M*f).symm (symm_infty _ hInf) hCi)
have hpconf : ∀ᵐ z : ℂ, IsConformalMap (fderiv ℝ (P : ℂ → ℂ) z) := by
filter_upwards [ae_conformal_post_biholomorphic f M hM hd] with z hz
exact hz.2
have hclass := weakQS_ae_conformal_affine_or_anti P hH hJ hP hPi hpconf
rcases restriction_affine_or_anti (M*f) hInf hclass with hleft|hright
· left
have hh := biholomorphic.mul_mem (biholomorphic.inv_mem hM) hleft
simpa only [← mul_assoc,inv_mul_cancel,one_mul] using hh
· right
have hm := reflection_conjugate_biholomorphic (biholomorphic.inv_mem hM)
have hh := biholomorphic.mul_mem hm hright
simp only [mul_assoc] at hh
rw [← mul_assoc boundaryReflection boundaryReflection, boundaryReflection_sq, one_mul,
← mul_assoc M⁻¹ M, inv_mul_cancel, one_mul] at hh
exact hh
end CannonMobius
end
end
noncomputable section
namespace CannonMobius
open Set Filter Topology MeasureTheory CannonMetricQC CannonConformal CannonQM CannonH3
open scoped OnePoint
variable {Γ : Type*} [Group Γ] [Countable Γ]
theorem sphere_action_straightening (A : Γ →* (Sphere ≃ₜ Sphere))
{η : Distortion} (hA : ∀ g, Controlled (A g) η) :
∃ F : Sphere ≃ₜ Sphere, ∀ g : Γ,
F*A g*F.symm ∈ biholomorphic ∨ boundaryReflection*(F*A g*F.symm) ∈ biholomorphic := by
obtain ⟨q,hqm,hqb,hqi,C,hC,F,hF0,hF1,hF,hFi,hD⟩ := controlled_action_structure_integration A hA
refine ⟨F.onePointCongr,fun g => ?_⟩
have hcr := weakQS_sphere_boundedCrossRatios F hC hF
have hcri : BoundedCrossRatios F.onePointCongr.symm := weakQS_sphere_boundedCrossRatios F.symm hC hFi
have hB : BoundedCrossRatios (F.onePointCongr*A g*F.onePointCongr.symm) :=
(hcr.comp (hA g).boundedCrossRatios (A g).injective).comp hcri F.onePointCongr.symm.injective
have hBi : BoundedCrossRatios (F.onePointCongr*A g*F.onePointCongr.symm).symm :=
(hcr.comp ((hA g).inverse (A g)).boundedCrossRatios (A g).symm.injective).comp
hcri F.onePointCongr.symm.injective
apply sphere_ae_conformal_rigidity _ hB hBi
exact integrated_conjugate_ae_conformal (A g) (hA g) q ((hqi g).mono (fun _ h => h.2))
F hF0 hC hF hFi hD
end CannonMobius
end
end
section
noncomputable section
section
open Set Topology
namespace CannonRelease
open Set Topology
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G)
variable (hthin : D.ThinTriangles)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open Set Topology
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) (hthin : D.ThinTriangles)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
open scoped NNReal
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology Set
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Filter Topology Set
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Set
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Set
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
namespace CannonRelease
open CannonGraph CannonVisual Set
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
local instance : Setoid (BasedRay D) := raySetoid D
end CannonRelease
end
end
end
end
end
end OAI