| import OAI.Analysis.CircleDomains.Topology.CompactScalarCircleModel |
|
|
| namespace OAI |
|
|
| /-! The empty-dust case uses the identity and the original Sobolev witnesses. -/ |
| open Set Function Filter MeasureTheory |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
|
|
| def compactScalarCircleModel_identity {m : ℕ} (u : Fin m → EuclideanPlane → ℝ) |
| (H : ∀ i, DeGiorgi.MemW1pWitness 2 (u i) univ) : |
| CompactScalarCircleModel univ u := by |
| have hf : extendAtInfinity (id : ℂ → ℂ) = (id : Sphere → Sphere) := by |
| funext p |
| cases p using OnePoint.rec <;> rfl |
| refine { |
| function := id |
| normalized := ?_ |
| conformal := ?_ |
| circle := ?_ |
| projection := id |
| projection_continuous := continuous_id |
| projection_inverse := ?_ |
| projection_outside := ?_ |
| projection_component := ?_ |
| witness := fun i => (H i).congr_ae (Eventually.of_forall fun x => ?_) |
| gradient_zero := ?_ } |
| · rw [hf] |
| refine ⟨rfl, ?_⟩ |
| simpa only [id_eq, finiteCoord_coe, sub_self] using |
| (Asymptotics.isBigO_zero (E' := ℂ) (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ)) |
| · simpa only [hf, image_id] using isConformalEquivalence_id (univ : Set Sphere) |
| · simpa only [hf, image_id] using isCircleDomain_univ |
| · intro p _ |
| rw [hf] |
| rfl |
| · intro p hp |
| exact (hp (by rw [hf, image_id]; trivial)).elim |
| · intro p hp |
| exact (hp (by rw [hf, image_id]; trivial)).elim |
| · simp only [id_eq, planeSphere, sphereFromPlaneFunction_coe, |
| ContinuousLinearEquiv.symm_apply_apply] |
| · intro i j |
| exact Eventually.of_forall fun x hx => (hx (by rw [hf, image_id]; trivial)).elim |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
