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a97875e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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