| import Mathlib |
| import OAI.Geometry.Cannon.Modulus.Ambient |
| import OAI.Geometry.Cannon.Conformal.Chart |
| import OAI.Geometry.Cannon.Modulus.SqDiffNormDiff |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace CannonRadialWitness |
| open Set Metric Topology CannonMobius |
| open scoped OnePoint |
|
|
| lemma trace_preconnected {S : Set Sphere} (hS : IsPreconnected S) (hinf : ∞∉S) : |
| IsPreconnected (((↑) : ℂ → Sphere) ⁻¹' S) := by |
| apply OnePoint.isOpenEmbedding_coe.isEmbedding.isInducing.isPreconnected_image.mp |
| have he : ((↑) : ℂ → Sphere) '' (((↑) : ℂ → Sphere) ⁻¹' S)=S := by |
| apply image_preimage_eq_of_subset |
| intro y hy |
| exact OnePoint.ne_infty_iff_exists.mp (fun he=>hinf (he ▸ hy)) |
| rwa [he] |
|
|
| theorem spherical_two_center_band {S : Set Sphere} (hS : IsPreconnected S) |
| {x p q : ℂ} {y : Sphere} (hx : (x:Sphere)∈S) (hy : y∈S) |
| {R η : ℝ} (hR : 0<R) (hpq : dist p q=R) (hxp : dist x p≤R) |
| (hη : 0≤η) (hη1 : η≤1) (hxy : 102*η*R^2<dist (x:Sphere) y^2) : |
| ∃ c : ℂ,(c=p ∨ c=q) ∧ ∃ a b : ℝ,0≤a ∧ a≤b ∧ b≤3*R ∧ b-a=η*R ∧ |
| ∃ u∈S,∃ v∈S,u∈((↑) '' closedBall c a : Set Sphere) ∧ |
| v∉((↑) '' ball c b : Set Sphere) := by |
| by_cases hinf : ∞∈S |
| · refine ⟨p,Or.inl rfl,R,R+η*R,hR.le,?_,?_,by ring,(x:Sphere),hx,∞,hinf,⟨x,hxp,rfl⟩,?_⟩ |
| · nlinarith |
| · nlinarith |
| · rintro ⟨z,_hz,he⟩ |
| exact OnePoint.coe_ne_infty z he |
| · obtain ⟨y',rfl⟩:=OnePoint.ne_infty_iff_exists.mp (show y≠∞ from fun he=>hinf (he ▸ hy)) |
| have hxy' : 102*η*R^2<dist x y'^2 := by |
| apply hxy.trans_le |
| exact pow_le_pow_left₀ dist_nonneg (coe_dist_le x y') 2 |
| obtain ⟨c,hc,a,b,ha,hab,hb,hw,u,hu,v,hv,hu',hv'⟩:=two_center_band_scaled |
| (trace_preconnected hS hinf) hx hy hR hpq hxp hη hη1 hxy' |
| refine ⟨c,hc,a,b,ha,hab,hb,hw,(u:Sphere),hu,(v:Sphere),hv,⟨u,hu',rfl⟩,?_⟩ |
| rintro ⟨z,hz,hez⟩ |
| have hzv : z=v:=OnePoint.coe_injective hez |
| subst z |
| exact not_lt_of_ge hv' hz |
|
|
| lemma continuum_point_at_radius {S : Set Sphere} (hS : IsPreconnected S) |
| {p : ℂ} (hp : (p:Sphere)∈S) (hinf : ∞∈S) {R : ℝ} (hR : 0≤R) : |
| ∃ q : ℂ,(q:Sphere)∈S ∧ dist p q=R := by |
| let f : C(Sphere,Sphere):=ContinuousMap.id Sphere |
| have h1 : f (p:Sphere)∈((↑) '' closedBall p R : Set Sphere):=⟨p,mem_closedBall_self hR,rfl⟩ |
| have h2 : f ∞∉((↑) '' ball p R : Set Sphere) := by |
| rintro ⟨z,_hz,he⟩ |
| exact OnePoint.coe_ne_infty z he |
| obtain ⟨x,hx,z,hz,hzx⟩:=CannonCarrierCharging.spherical_radial_level_preimage f S hS p hp hinf h1 h2 R ⟨le_rfl,le_rfl⟩ |
| refine ⟨z,?_,?_⟩ |
| · exact hzx ▸ hx |
| · simpa only [mem_sphere,dist_comm] using hz |
| end CannonRadialWitness |
|
|
| end |
| end |
|
|
| end OAI |
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