| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Minimizers.Minimum |
| import OAI.Geometry.CAT0Fillings.Radial.ClosedInequality |
| import OAI.Geometry.CAT0Fillings.Powers.PolynomialBound |
| import OAI.Geometry.CAT0Fillings.Euler.Contradiction |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| open ChartGeometry AnalyticMinimizer MassMeasure Rearrangement RadialSobolev |
|
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| theorem extremal_euler_powers {k : ℕ} (hk : 0 < k) (hX : IsCAT0 X) |
| {T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : boundarySucc T = 0) |
| (hm : 0 < mass T) (q : ChartGeometry hT.1) |
| {d r : ℝ} (hd : 0 < d) (hr : 1 < r) |
| (hfill : ∀ P : Functional X (k+1), IsIntegral (k+1) P → boundarySucc P = 0 → |
| ∃ R : Functional X (k+2), IsIntegral (k+2) R ∧ boundarySucc R = P ∧ |
| mass R ≤ fillingCoefficient (k+1)*(mass P)^(fillingPower (k+1))) |
| (hext : ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 → |
| d*((mass T)^r-(mass B)^r) ≤ fillingVolume (T-B)) |
| (v : q.Sobolev) |
| (hv : ∀ᵐ x ∂currentMassMeasure hT.1, 0 ≤ q.inclusion v x) |
| (heuler : ∀ ψ : q.Sobolev, |
| (4/((k+2:ℝ)-2))*inner ℝ (q.closedGradient v) (q.closedGradient ψ) + |
| (k+2:ℝ)*inner ℝ (q.inclusion v) (q.inclusion ψ) = |
| (k+2:ℝ)*(∫ x, ((q.inclusion v) x)^(sobolevP (k+2)-1)*(q.inclusion ψ) x ∂currentMassMeasure hT.1)) : |
| (∀ b : ℝ, 0 < b → MemLp (q.inclusion v) (ENNReal.ofReal b) (currentMassMeasure hT.1)) ∧ |
| (∀ γ : ℝ, 1 ≤ γ → ∃ R : q.Sobolev, |
| (q.inclusion R : X → ℝ) =ᵐ[currentMassMeasure hT.1] (fun x => (q.inclusion v x)^γ) ∧ |
| (q.closedGradient R : _ → _) =ᵐ[q.atlasMeasure] |
| (fun w => (γ*(q.inclusion v (q.atlasParam w))^(γ-1)) • q.closedGradient v w)) := by |
| have hn : (2:ℝ) < k+2 := by exact_mod_cast (show 2 < k+2 by omega) |
| have hk0 : 0 < (k+2:ℝ)-2 := by linarith |
| have hp : 2 < sobolevP (k+2) := by |
| dsimp [sobolevP] |
| push_cast |
| exact (lt_div_iff₀ hk0).mpr (by linarith) |
| have hB : 0 < (k+2:ℝ) := by positivity |
| have hsphere : 0 < sphereArea (k+2) := by |
| unfold sphereArea |
| exact mul_pos (by positivity) (omega_pos (by omega)) |
| let S := (k+2:ℝ)*sphereArea (k+2)^(2/(k+2:ℝ)) |
| have hS : 0 < S := mul_pos hB (Real.rpow_pos_of_pos hsphere _) |
| obtain ⟨C,hC,hb⟩ := extremal_closed_sobolev hk hX hT hz hm q hd hr hfill hext |
| (show 0 < S/2 by positivity) (show S/2 < S by linarith) |
| exact q.closed_all_powers hz v hp (div_pos (by norm_num) hk0) hB |
| (show 0 < S-S/2 by linarith) hC hv hb heuler |
|
|
| end CAT0Fillings |
| end |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| open ChartGeometry AnalyticMinimizer MassMeasure Rearrangement RadialSobolev |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
| theorem normalized_extremal_contradiction {k : ℕ} (hk : 0 < k) (hX : IsCAT0 X) |
| {T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : boundarySucc T = 0) |
| (hm : 0 < mass T) (hms : mass T < sphereArea (k+2)) |
| {d r : ℝ} (hd : 0 < d) (hr : 1 < r) |
| (hnorm : d*r*(mass T)^(r-1) = 1/(k+2:ℝ)) |
| (hfill : ∀ P : Functional X (k+1), IsIntegral (k+1) P → boundarySucc P = 0 → |
| ∃ R : Functional X (k+2), IsIntegral (k+2) R ∧ boundarySucc R = P ∧ |
| mass R ≤ fillingCoefficient (k+1)*(mass P)^(fillingPower (k+1))) |
| (hext : ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 → |
| d*((mass T)^r-(mass B)^r) ≤ fillingVolume (T-B)) : False := by |
| obtain ⟨q⟩ := exists_chartGeometry hX hT.1 hT.2.1 |
| obtain ⟨v,hv,hW,hWs,hE,heuler,hmin⟩ := extremal_normalized_minimum hk hX hT hz hm hms q hd hr hfill hext |
| obtain ⟨hmom,hpow⟩ := extremal_euler_powers hk hX hT hz hm q hd hr hfill hext v hv heuler |
| let β : ℝ := 1/((k+2:ℝ)-2) |
| have hk0 : 0 < (k:ℝ) := Nat.cast_pos.mpr hk |
| have hb : 0 < β := by dsimp [β]; exact div_pos (by norm_num) (by linarith) |
| have hnb : (k+2:ℝ)*β = 1+2*β := by dsimp [β]; simp only [add_sub_cancel_right]; field_simp [hk0.ne'] |
| have hp : sobolevP (k+2) = 2+4*β := by |
| dsimp [sobolevP,β] |
| push_cast |
| simp only [add_sub_cancel_right] |
| field_simp [hk0.ne'] |
| ring |
| have ha : 4/((k+2:ℝ)-2) = 4*β := by dsimp [β]; ring |
| have hψ : ∀ ψ : q.Sobolev, |
| 4*β*inner ℝ (q.closedGradient v) (q.closedGradient ψ) + |
| (k+2:ℝ)*inner ℝ (q.inclusion v) (q.inclusion ψ) = |
| (k+2:ℝ)*(∫ x, ((q.inclusion v) x)^(1+4*β)*(q.inclusion ψ) x ∂currentMassMeasure hT.1) := by |
| intro ψ |
| simpa only [hp,ha,show 2+4*β-1 = 1+4*β by ring] using heuler ψ |
| have hrad (o : X) (g : X → ℝ) (K : ℝ≥0) (hg : LipschitzWith K g) |
| (hg0 : ∀ x, 0 ≤ g x) (hg1 : ∀ x, g x ≤ 1) : |
| q.radialVariation o g ≤ (k+2:ℝ)*q.sweptMass o g := by |
| have h := radial_inequality_of_extremal hX hT hz q hd.le (by linarith : 0 ≤ r) hm hext o hg hg0 hg1 |
| rw [hnorm] at h |
| have hh := mul_le_mul_of_nonneg_left h (show 0 ≤ (k+2:ℝ) by positivity) |
| simpa only [←mul_assoc,mul_one_div_cancel (show (k+2:ℝ) ≠ 0 by positivity),one_mul] using hh |
| apply q.euler_chord_contradiction hX (by omega) hz v hb ?_ hv hmom hpow ?_ ?_ ?_ ?_ ?_ ?_ |
| · simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hnb |
| · simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hψ |
| · simpa only [Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hrad |
| · rwa [hp] at hW |
| · rwa [hp] at hWs |
| · simpa only [hp,ha,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hE |
| · simpa only [hp,ha,Nat.cast_add,Nat.cast_one,add_assoc,show (1:ℝ)+1=2 by norm_num] using hmin |
| end CAT0Fillings |
| end |
|
|
| end OAI |
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