| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Variational.Ekeland |
| import OAI.Geometry.CAT0Fillings.Variational.LowerContinuity |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings |
| open Foundations Slicing CurrentOperations |
|
|
| lemma exists_mass_bound_of_positive_deficit {A c p : ℝ} (hc : 0 < c) (hp : 1 < p) : |
| ∃ M : ℝ≥0, ∀ m : ℝ, 0 ≤ m → 0 ≤ A*m-c*m^p → m ≤ M := by |
| obtain ⟨R,hR⟩ := eventually_atTop.mp ((tendsto_rpow_atTop (sub_pos.mpr hp)).eventually |
| (eventually_ge_atTop ((A+1)/c))) |
| refine ⟨⟨max R 1,le_trans zero_le_one (le_max_right _ _)⟩,fun m hm hpos => ?_⟩ |
| by_contra hnot |
| have hlarge : max R 1 < m := lt_of_not_ge hnot |
| have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one ((le_max_right R 1).trans hlarge.le) |
| have hh := hR m ((le_max_left R 1).trans hlarge.le) |
| have hpow : m^p = m^(p-1)*m := by |
| conv_lhs => rw [show p = (p-1)+1 by ring] |
| rw [Real.rpow_add hm0,Real.rpow_one] |
| have hmul : A+1 ≤ c*m^(p-1) := by |
| exact (div_le_iff₀ hc).mp hh |>.trans_eq (mul_comm _ _) |
| rw [hpow] at hpos |
| nlinarith |
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
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| noncomputable def fillingDeficit {k : ℕ} (c p : ℝ) (T : Functional X (k+1)) : ℝ := |
| fillingVolume T-c*(mass T)^p |
|
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| theorem exists_filling_extremizer (hX : IsCAT0 X) {k : ℕ} {c p δ : ℝ} |
| (hc : 0 < c) (hp : 1 < p) (hδ : 0 < δ) |
| {T0 : Functional X (k+2)} (hT0 : IsIntegral (k+2) T0) (hz0 : boundarySucc T0 = 0) |
| (hpositive : 0 < fillingDeficit c p T0) : |
| ∃ T : Functional X (k+2), IsIntegral (k+2) T ∧ boundarySucc T = 0 ∧ |
| 0 < mass T ∧ fillingDeficit c p T0 ≤ fillingDeficit c p T ∧ |
| ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 → |
| c/(1+δ)*((mass T)^p-(mass B)^p) ≤ fillingVolume (T-B) := by |
| classical |
| have hpos : 0 < fillingVolume T0-c*(mass T0)^p := hpositive |
| let A : ℝ := (k+3:ℝ)*diam (univ : Set X) |
| have hA : 0 ≤ A := mul_nonneg (by positivity) diam_nonneg |
| obtain ⟨M,hM⟩ := exists_mass_bound_of_positive_deficit (A := A) hc hp |
| let C := {T : Functional X (k+2) // IsIntegral (k+2) T ∧ boundarySucc T = 0 ∧ |
| fillingDeficit c p T0 ≤ fillingDeficit c p T} |
| let F : C → ℝ := fun T => fillingDeficit c p T.val |
| let d : C → C → ℝ := fun T U => fillingVolume (T.val-U.val) |
| have hlin (T : C) : fillingVolume T.val ≤ A*mass T.val := by |
| simpa only [A,Nat.cast_add,Nat.cast_one,show (1:ℝ)+2 = 3 by norm_num,add_assoc] using |
| hX.fillingVolume_le_linear T.property.1 T.property.2.1 |
| have hmass (T : C) : mass T.val ≤ M := by |
| apply hM _ (mass_nonneg _) |
| have hh := T.property.2.2 |
| have hl := hlin T |
| dsimp only [fillingDeficit] at hh |
| linarith |
| have hbound : BddAbove (range F) := by |
| refine ⟨A*M,?_⟩ |
| rintro v ⟨T,rfl⟩ |
| have hl := hlin T |
| have hm := mul_le_mul_of_nonneg_left (hmass T) hA |
| have hn := mul_nonneg hc.le (Real.rpow_nonneg (mass_nonneg T.val) p) |
| dsimp only [F,fillingDeficit] |
| linarith |
| have hcluster : ∀ u : ℕ → C, ∀ L : ℝ, |
| Tendsto (fun j => F (u j)) atTop (𝓝 L) → |
| (∀ ε > 0, ∃ N : ℕ, ∀ i ≥ N, ∀ j ≥ N, d (u i) (u j) < ε) → |
| ∃ x : C, Tendsto (fun j => d (u j) x) atTop (𝓝 0) ∧ L ≤ F x := by |
| intro u L hL hdc |
| obtain ⟨m,hmI,σ,hσ,hm⟩ := isCompact_Icc.isSeqCompact |
| (fun j => (show mass (u j).val ∈ Icc (0:ℝ) M from ⟨mass_nonneg _,hmass _⟩)) |
| obtain ⟨ψ,hψ,T,hT,hTM,_,hweak⟩ := exists_integral_weak_subsequence |
| (fun j => (u (σ j)).property.1) M 0 (fun j => hmass _) |
| (fun j => by |
| rw [(u (σ j)).property.2.1] |
| change mass (fun _ _ => (0:ℝ)) ≤ (0:ℝ) |
| rw [mass_zero]) hX |
| let τ : ℕ → ℕ := σ ∘ ψ |
| have hτ : StrictMono τ := hσ.comp hψ |
| have hz : boundarySucc T = 0 := by |
| funext b π |
| apply tendsto_nhds_unique (boundarySucc_weak_limit hweak b π) |
| simpa only [(u _).property.2.1,Pi.zero_apply] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) |
| have hdlim : Tendsto (fun j => fillingVolume ((u j).val-T)) atTop (𝓝 0) := |
| fillingVolume_sub_tendsto_zero hX (fun j => (u j).property.1) |
| (fun j => (u j).property.2.1) M (fun j => hmass _) hdc hτ hweak |
| have hV := fillingVolume_tendsto_of_sub_tendsto_zero hX |
| (fun j => (u j).property.1) (fun j => (u j).property.2.1) hT hz hdlim |
| have hmlim : Tendsto (fun j => mass (u (τ j)).val) atTop (𝓝 m) := hm.comp hψ.tendsto_atTop |
| have hmle : mass T ≤ m := mass_le_of_mass_tendsto |
| (fun j => (u (τ j)).property.1.1) hmlim hweak |
| have hFlim : Tendsto (fun j => F (u (τ j))) atTop (𝓝 (fillingVolume T-c*m^p)) := |
| (hV.comp hτ.tendsto_atTop).sub ((hmlim.rpow_const (Or.inr (by linarith))).const_mul c) |
| have heq : L = fillingVolume T-c*m^p := tendsto_nhds_unique |
| (hL.comp hτ.tendsto_atTop) hFlim |
| have hLle : L ≤ fillingDeficit c p T := by |
| rw [heq] |
| have hpow := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow (mass_nonneg T) hmle (by linarith : 0 ≤ p)) hc.le |
| dsimp only [fillingDeficit] |
| linarith |
| have hT0le : fillingDeficit c p T0 ≤ L := ge_of_tendsto hL |
| (Eventually.of_forall fun j => (u j).property.2.2) |
| exact ⟨⟨T,hT,hz,hT0le.trans hLle⟩,hdlim,hLle⟩ |
| let x0 : C := ⟨T0,hT0,hz0,le_refl _⟩ |
| obtain ⟨x,hx0,hx⟩ := exists_variational_point d |
| (fun x => by dsimp only [d]; rw [sub_self,fillingVolume_zero]) |
| (fun x y => fillingVolume_nonneg _) |
| (fun x y => fillingVolume_sub_symm hX x.property.1 x.property.2.1 y.property.1 y.property.2.1) |
| (fun x y z => fillingVolume_triangle hX x.property.1 x.property.2.1 |
| y.property.1 y.property.2.1 z.property.1 z.property.2.1) |
| F hbound hcluster x0 hδ |
| have hx0' : fillingDeficit c p T0 ≤ fillingDeficit c p x.val := hx0 |
| refine ⟨x.val,x.property.1,x.property.2.1,?_,hx0',?_⟩ |
| · by_contra hnot |
| have hm0 : mass x.val = 0 := le_antisymm (le_of_not_gt hnot) (mass_nonneg _) |
| have hl := hlin x |
| have hp0 : p ≠ 0 := by linarith |
| simp only [hm0,Real.zero_rpow hp0,mul_zero,fillingDeficit] at hl hx0' |
| linarith |
| · intro B hB hzB |
| have hF : fillingDeficit c p B ≤ fillingDeficit c p x.val+δ*fillingVolume (x.val-B) := by |
| by_cases hb : fillingDeficit c p T0 ≤ fillingDeficit c p B |
| · exact hx ⟨B,hB,hzB,hb⟩ |
| · have hn := mul_nonneg hδ.le (fillingVolume_nonneg (x.val-B)) |
| linarith [lt_of_not_ge hb] |
| have hV := (le_abs_self (fillingVolume x.val-fillingVolume B)).trans |
| (abs_fillingVolume_sub_le hX x.property.1 x.property.2.1 hB hzB) |
| have hh : c*((mass x.val)^p-(mass B)^p) ≤ (1+δ)*fillingVolume (x.val-B) := by |
| dsimp only [fillingDeficit] at hF |
| nlinarith |
| have hden : 0 < 1+δ := by linarith |
| rw [div_mul_eq_mul_div,div_le_iff₀ hden] |
| nlinarith |
|
|
| end CAT0Fillings |
| end |
|
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| end OAI |
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