| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Charts.Quadratic |
| import OAI.Geometry.CAT0Fillings.Differentiation.LocalBounds |
| import OAI.Geometry.CAT0Fillings.Charts.JacobianBound |
| import OAI.Geometry.CAT0Fillings.Charts.CurrentContinuity |
| import OAI.Geometry.CAT0Fillings.Charts.ScalarPieces |
| import OAI.Geometry.CAT0Fillings.Slicing.CycleCoarea |
| import OAI.Geometry.CAT0Fillings.Currents.IntegralPush |
| import OAI.Geometry.CAT0Fillings.Prism.Differential |
| import OAI.Geometry.CAT0Fillings.Prism.Action |
| import OAI.Geometry.CAT0Fillings.Currents.CycleDerivative |
| import OAI.Geometry.CAT0Fillings.Currents.CycleFTC |
| import OAI.Geometry.CAT0Fillings.Fillings.Existence |
| import OAI.Geometry.CAT0Fillings.Fillings.ConeBound |
| import OAI.Geometry.CAT0Fillings.Geometry.DeterminantBound |
| import OAI.Geometry.CAT0Fillings.Radial.FirstVariation |
| import OAI.Geometry.CAT0Fillings.Tangent.Transfer |
| import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain |
| import OAI.Geometry.CAT0Fillings.Calculus.ClosedLevel |
| import OAI.Geometry.CAT0Fillings.Calculus.VectorLimit |
| import OAI.Geometry.CAT0Fillings.Calculus.Piecewise |
| import OAI.Geometry.CAT0Fillings.Calculus.Truncation |
| import OAI.Geometry.CAT0Fillings.Calculus.PositiveBivariate |
| import OAI.Geometry.CAT0Fillings.Powers.ClosedLimit |
| import OAI.Geometry.CAT0Fillings.Calculus.Polynomial |
| import OAI.Geometry.CAT0Fillings.Compactness.Projected |
| import OAI.Geometry.CAT0Fillings.Compactness.Borel |
| import OAI.Geometry.CAT0Fillings.Compactness.Interpolation |
| import OAI.Geometry.CAT0Fillings.Mass.Gradient |
| import OAI.Geometry.CAT0Fillings.Rearrangement.Gradient |
| import OAI.Geometry.CAT0Fillings.Minimizers.Nonnegative |
| import OAI.Geometry.CAT0Fillings.Chord.GroundState |
| import OAI.Geometry.CAT0Fillings.Chord.GroundProducts |
| import OAI.Geometry.CAT0Fillings.Calculus.Atlas |
| import OAI.Geometry.CAT0Fillings.Calculus.ChainSet |
| import OAI.Geometry.CAT0Fillings.Radial.Extension |
| import OAI.Geometry.CAT0Fillings.Chord.RadialIntegral |
| import OAI.Geometry.CAT0Fillings.Chord.TestIdentity |
| import OAI.Geometry.CAT0Fillings.Chord.Distribution |
| import OAI.Geometry.CAT0Fillings.Chord.DensityContradiction |
| import OAI.Geometry.CAT0Fillings.Euler.OppositeIntegral |
| import OAI.Geometry.CAT0Fillings.Euler.Contradiction |
| import OAI.Geometry.CAT0Fillings.Density.NearEuclidean |
| import OAI.Geometry.CAT0Fillings.Density.Normalized |
| import OAI.Geometry.CAT0Fillings.Fillings.Compact |
| import OAI.Geometry.CAT0Fillings.Geometry.Ball |
| import OAI.Geometry.CAT0Fillings.Transport.Proper |
| import OAI.Geometry.CAT0Fillings.Transport.Support |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| open Foundations CurrentOperations |
|
|
| universe u |
| variable {X : Type u} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
| lemma proper_filling_bound [Nonempty X] (hX : IsCAT0 X) {k : ℕ} |
| {T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hTc : CompactlySupported T) |
| (hz : boundarySucc T = 0) : |
| ∃ S : Functional X (k+3), IsIntegral (k+3) S ∧ CompactlySupported S ∧ |
| boundarySucc S = T ∧ mass S ≤ fillingCoefficient (k+2)*(mass T)^(fillingPower (k+2)) := by |
| classical |
| obtain ⟨K,hK,hTK⟩ := hTc |
| let o : X := Classical.choice inferInstance |
| obtain ⟨R,hR,hKR⟩ := hK.isBounded.subset_closedBall_lt 0 o |
| obtain ⟨P,hP,hPr,hPf⟩ := hX.exists_ball_retraction o (R := R+2) (by linarith) |
| let B := closedBall o (R+2) |
| have : Nonempty B := ⟨o,by simp [B]; linarith⟩ |
| let p : X → B := fun x => ⟨P x,by simpa only [B,mem_closedBall,dist_comm] using hPr x⟩ |
| have hp : LipschitzWith 1 p := hP |
| have hinc : LipschitzWith 1 (Subtype.val : B → X) := LipschitzWith.subtype_val B |
| have hiA : AntilipschitzWith 1 (Subtype.val : B → X) := by |
| intro x y |
| change edist (x : X) (y : X) ≤ (1:ℝ≥0∞)*edist (x : X) (y : X) |
| rw [one_mul] |
| let U := pushCurrent p T |
| have hU : IsIntegral (k+2) U := integral_push_proper hT hp |
| have hUz : boundarySucc U = 0 := pushCurrent_cycle hz hp |
| have hUm : mass U ≤ mass T := mass_push_one_general hT.1 hp |
| have hB : IsCAT0 B := hX.closedBall o (by linarith) |
| obtain ⟨V,hV,hVb,hVm⟩ := hB.exists_minimal_integral_filling hU hUz |
| let S := pushCurrent (Subtype.val : B → X) V |
| have hS : IsIntegral (k+3) S := integral_push_of_bilipschitz hV hinc hiA |
| have hb : boundarySucc S = T := by |
| rw [pushCurrent_boundarySucc V hinc,hVb,←pushCurrent_comp hinc] |
| exact pushCurrent_eq_of_ball_support hT.1 hTK o R hKR hP |
| (fun x hx => hPf x (by rwa [dist_comm])) |
| refine ⟨S,hS,?_,hb,?_⟩ |
| · refine ⟨B,isCompact_closedBall o (R+2),?_⟩ |
| intro b π hab hbK |
| change pushCurrent (Subtype.val : B → X) V b π = 0 |
| rw [pushCurrent_apply _ _ hab] |
| have heq : b ∘ (Subtype.val : B → X) = fun _ => 0 := funext fun x => hbK x x.property |
| rw [heq] |
| have hπ := (admissible_comp hab hinc).2 |
| simpa using hV.1.linearFirst (fun _ => 0) (fun _ => 0) (fun i => π i ∘ Subtype.val) |
| 0 0 (BoundedLip.const 0) (BoundedLip.const 0) hπ |
| · calc |
| mass S ≤ mass V := mass_push_one_general hV.1 hinc |
| _ = fillingVolume U := hVm |
| _ ≤ fillingCoefficient (k+2)*(mass U)^(fillingPower (k+2)) := compact_filling_bound k B hB U hU hUz |
| _ ≤ fillingCoefficient (k+2)*(mass T)^(fillingPower (k+2)) := |
| mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (mass_nonneg U) hUm |
| (le_of_lt (lt_trans zero_lt_one (fillingPower_gt_one (by omega))))) |
| (fillingCoefficient_pos (by omega)).le |
|
|
| theorem sharp_integral_filling |
| (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
| (hX : IsCAT0 X) (n : ℕ) (hn : 2 ≤ n) (T : IntegralCurrent X n) |
| (hTc : CompactlySupported T.val) (hTz : IsCycle T.val) : |
| ∃ S : IntegralCurrent X (n + 1), CompactlySupported S.val ∧ |
| boundarySucc S.val = T.val ∧ |
| mass S.val ≤ fillingCoefficient n * (mass T.val) ^ fillingPower n := by |
| classical |
| obtain ⟨k,hk⟩ := Nat.exists_eq_add_of_le hn |
| have hnEq : n = k+2 := by omega |
| clear hk |
| subst n |
| rcases isEmpty_or_nonempty X with hE | hN |
| · have hT0 : T.val = 0 := by |
| funext b π |
| by_cases hab : Admissible b π |
| · have heq : b = fun _ => 0 := funext fun x => isEmptyElim x |
| rw [heq] |
| simpa using T.property.1.linearFirst (fun _ => 0) (fun _ => 0) π 0 0 |
| (BoundedLip.const 0) (BoundedLip.const 0) hab.2 |
| · exact T.property.1.offDomain b π hab |
| refine ⟨⟨0,isIntegral_zero _⟩,compactlySupported_zero _,?_,?_⟩ |
| · rw [hT0]; exact boundarySucc_zero _ |
| · rw [show mass (0 : Functional X (k+2+1)) = 0 from mass_zero _] |
| exact mul_nonneg (fillingCoefficient_pos (by omega)).le (Real.rpow_nonneg (mass_nonneg _) _) |
| · obtain ⟨S,hS,hSc,hSb,hSm⟩ := proper_filling_bound hX T.property hTc hTz |
| exact ⟨⟨S,hS⟩,hSc,hSb,hSm⟩ |
| end CAT0Fillings |
| end |
|
|
| end OAI |
|
|
