| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.PushBoundary |
| import OAI.Geometry.CAT0Fillings.Charts.Postcompose |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory |
| open CurrentOperations |
|
|
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] |
| [MeasurableSpace X] [MeasurableSpace Y] [BorelSpace X] [BorelSpace Y] |
| [CompactSpace X] |
| lemma integerRectifiable_push_of_bilipschitz {k : ℕ} {T : Functional X k} |
| (hT : IntegerRectifiable T) {f : X → Y} {K J : ℝ≥0} |
| (hf : LipschitzWith K f) (hfJ : AntilipschitzWith J f) : |
| IntegerRectifiable (pushCurrent f T) := by |
| classical |
| obtain ⟨C,hdis,hC,hm,heq⟩ := hT |
| choose L U hL hU using fun i => (C i).bilipschitz |
| let D : ℕ → IntegerChart Y k := fun i => (C i).postcompose hf (hfJ.comp (hU i)) |
| have hDaction (i : ℕ) : (D i).action = pushCurrent f (C i).action := |
| (C i).postcompose_action hf (hfJ.comp (hU i)) |
| have hD (i : ℕ) : IsMetricCurrent (D i).action := by |
| rw [hDaction] |
| exact pushCurrent_isMetricCurrent (hC i) hf |
| refine ⟨D,?_,hD,?_,?_⟩ |
| · intro i j hij |
| change Disjoint ((C i).postcompose hf _).image ((C j).postcompose hf _).image |
| rw [(C i).postcompose_image,(C j).postcompose_image] |
| exact (Set.disjoint_image_iff hfJ.injective).mpr (hdis hij) |
| · apply (hm.mul_left ((K : ℝ)^k)).of_nonneg_of_le |
| · intro i |
| exact mass_nonneg (D i).action |
| · intro i |
| rw [hDaction] |
| exact mass_pushCurrent_le (hC i) hf |
| · intro b π |
| by_cases h : Admissible b π |
| · rw [pushCurrent_apply f T h,heq] |
| exact tsum_congr (fun i => by rw [hDaction,pushCurrent_apply f (C i).action h]) |
| · rw [show pushCurrent f T b π = 0 from ite_eq_right h] |
| symm |
| calc |
| (∑' i, (D i).action b π) = ∑' (_i : ℕ), (0 : ℝ) := |
| tsum_congr (fun i => (hD i).offDomain b π h) |
| _ = 0 := tsum_zero |
|
|
| lemma integral_push_of_bilipschitz {k : ℕ} {T : Functional X k} |
| (hT : IsIntegral k T) {f : X → Y} {K J : ℝ≥0} |
| (hf : LipschitzWith K f) (hfJ : AntilipschitzWith J f) : |
| IsIntegral k (pushCurrent f T) := by |
| cases k with |
| | zero => exact ⟨pushCurrent_isMetricCurrent hT.1 hf, |
| integerRectifiable_push_of_bilipschitz hT.2 hf hfJ⟩ |
| | succ k => |
| refine ⟨pushCurrent_isMetricCurrent hT.1 hf, |
| integerRectifiable_push_of_bilipschitz hT.2.1 hf hfJ,?_,?_⟩ |
| · rw [pushCurrent_boundarySucc T hf] |
| exact pushCurrent_isMetricCurrent hT.2.2.1 hf |
| · rw [pushCurrent_boundarySucc T hf] |
| exact integerRectifiable_push_of_bilipschitz hT.2.2.2 hf hfJ |
|
|
| end CAT0Fillings |
|
|
| end |
| end |
|
|
| end OAI |
|
|
