| import Mathlib |
| import OAI.Geometry.IntegralFillings.Slicing.EuclideanSplit |
|
|
| namespace OAI |
|
|
| section |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| namespace SharpIntegralFillings.IntegerChart |
| open Set MeasureTheory Filter |
| open scoped Topology |
|
|
| variable {X : Type*} [MetricSpace X] {k : ℕ} (C : IntegerChart X (k+1)) |
| noncomputable def coordinateLayer (k : ℕ) (t : ℝ) (z : Euc k) : Euc (k+1) := |
| (Prism.split k).symm (t,z) |
|
|
| lemma coordinateLayer_lipschitz (t : ℝ) : |
| LipschitzWith ‖(Prism.split k).symm.toContinuousLinearMap‖₊ (coordinateLayer k t) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro z w |
| have h := (Prism.split k).symm.lipschitzWith.dist_le_mul (t,z) (t,w) |
| simpa only [coordinateLayer,Prod.dist_eq,dist_self,max_eq_right dist_nonneg] using h |
|
|
| lemma coordinateLayer_antilipschitz (t : ℝ) : |
| AntilipschitzWith ‖(Prism.split k).toContinuousLinearMap‖₊ (coordinateLayer k t) := by |
| apply AntilipschitzWith.of_le_mul_dist |
| intro z w |
| have h := (Prism.split k).lipschitzWith.dist_le_mul |
| (coordinateLayer k t z) (coordinateLayer k t w) |
| simpa only [coordinateLayer,ContinuousLinearEquiv.apply_symm_apply,Prod.dist_eq,dist_self, |
| max_eq_right dist_nonneg] using h |
|
|
| noncomputable def coordinateDomain (t : ℝ) : Set (Euc k) := |
| coordinateLayer k t ⁻¹' C.domain |
| |
| lemma measurableSet_coordinateDomain (t : ℝ) : MeasurableSet (C.coordinateDomain t) := |
| C.borel.preimage (coordinateLayer_lipschitz (k := k) t).continuous.measurable |
| |
| lemma bounded_coordinateDomain (t : ℝ) : Bornology.IsBounded (C.coordinateDomain t) := by |
| apply Bornology.IsBounded.subset |
| ((LipschitzWith.prod_snd : LipschitzWith 1 (Prod.snd : ℝ × Euc k → Euc k)).isBounded_image |
| ((Prism.split k).lipschitzWith.isBounded_image C.bounded)) |
| intro z hz |
| refine ⟨(t,z),?_,rfl⟩ |
| exact ⟨coordinateLayer k t z,hz,(Prism.split k).apply_symm_apply (t,z)⟩ |
| |
| noncomputable def coordinateParam (t : ℝ) (z : C.coordinateDomain t) : X := |
| C.param ⟨coordinateLayer k t z,z.2⟩ |
| |
| lemma coordinateParam_lipschitz (t : ℝ) {L : ℝ≥0} (hL : LipschitzWith L C.param) : |
| LipschitzWith (L * ‖(Prism.split k).symm.toContinuousLinearMap‖₊) (C.coordinateParam t) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro z w |
| have h := hL.dist_le_mul ⟨coordinateLayer k t z,z.2⟩ ⟨coordinateLayer k t w,w.2⟩ |
| have hh := (coordinateLayer_lipschitz (k := k) t).dist_le_mul z.val w.val |
| change dist (C.coordinateParam t z) (C.coordinateParam t w) ≤ _ at h |
| calc |
| _ ≤ (L:ℝ) * dist (coordinateLayer k t z) (coordinateLayer k t w) := h |
| _ ≤ (L:ℝ) * (‖(Prism.split k).symm.toContinuousLinearMap‖₊ * dist z.val w.val) := |
| mul_le_mul_of_nonneg_left hh L.coe_nonneg |
| _ = _ := by simp only [NNReal.coe_mul,mul_assoc,Subtype.dist_eq] |
| |
| lemma coordinateParam_antilipschitz (t : ℝ) {U : ℝ≥0} (hU : AntilipschitzWith U C.param) : |
| AntilipschitzWith (‖(Prism.split k).toContinuousLinearMap‖₊ * U) (C.coordinateParam t) := by |
| apply AntilipschitzWith.of_le_mul_dist |
| intro z w |
| have h := hU.le_mul_dist ⟨coordinateLayer k t z,z.2⟩ ⟨coordinateLayer k t w,w.2⟩ |
| have hh := (coordinateLayer_antilipschitz (k := k) t).le_mul_dist z.val w.val |
| calc |
| dist z w ≤ ‖(Prism.split k).toContinuousLinearMap‖₊ * |
| dist (coordinateLayer k t z) (coordinateLayer k t w) := hh |
| _ ≤ ‖(Prism.split k).toContinuousLinearMap‖₊ * |
| ((U:ℝ)*dist (C.coordinateParam t z) (C.coordinateParam t w)) := |
| mul_le_mul_of_nonneg_left h (NNReal.coe_nonneg _) |
| _ = _ := by simp only [NNReal.coe_mul,mul_assoc] |
| |
| lemma coordinateParam_bilipschitz (t : ℝ) : ∃ L U : ℝ≥0, |
| LipschitzWith L (C.coordinateParam t) ∧ AntilipschitzWith U (C.coordinateParam t) := by |
| obtain ⟨L,U,hL,hU⟩ := C.bilipschitz |
| exact ⟨_,_,C.coordinateParam_lipschitz t hL,C.coordinateParam_antilipschitz t hU⟩ |
| |
| lemma ae_integrable_coordinateMultiplicity : ∀ᵐ t : ℝ, |
| Integrable (fun z => (C.multiplicity (coordinateLayer k t z):ℝ)) |
| (volume.restrict (C.coordinateDomain t)) := by |
| have hi : Integrable (C.domain.indicator (fun z => (C.multiplicity z:ℝ))) := |
| (integrable_indicator_iff C.borel).mpr C.integrable |
| have hip : Integrable (fun p : ℝ × Euc k => |
| C.domain.indicator (fun z => (C.multiplicity z:ℝ)) ((Prism.split k).symm p)) := |
| ((Prism.split_symm_measurePreserving k).integrable_comp_emb |
| (Prism.split k).symm.toHomeomorph.measurableEmbedding).mpr hi |
| filter_upwards [hip.prod_right_ae] with t ht |
| apply (integrable_indicator_iff (C.measurableSet_coordinateDomain t)).mp |
| have heq : (C.coordinateDomain t).indicator (fun z => (C.multiplicity (coordinateLayer k t z):ℝ)) = |
| fun z => C.domain.indicator (fun w => (C.multiplicity w:ℝ)) ((Prism.split k).symm (t,z)) := by |
| funext z |
| change (C.coordinateDomain t).indicator _ z = C.domain.indicator _ (coordinateLayer k t z) |
| by_cases hz : coordinateLayer k t z ∈ C.domain |
| · rw [indicator_of_mem (show z ∈ C.coordinateDomain t from hz),indicator_of_mem hz] |
| · rw [indicator_of_notMem (show z ∉ C.coordinateDomain t from hz),indicator_of_notMem hz] |
| rw [heq] |
| exact ht |
| |
| noncomputable def coordinateSlice (t : ℝ) |
| (ht : Integrable (fun z => (C.multiplicity (coordinateLayer k t z):ℝ)) |
| (volume.restrict (C.coordinateDomain t))) : IntegerChart X k where |
| domain := C.coordinateDomain t |
| borel := C.measurableSet_coordinateDomain t |
| bounded := C.bounded_coordinateDomain t |
| param := C.coordinateParam t |
| bilipschitz := C.coordinateParam_bilipschitz t |
| multiplicity z := C.multiplicity (coordinateLayer k t z) |
| integrable := ht |
| |
| end SharpIntegralFillings.IntegerChart |
| |
| end |
| |
| end OAI |
| |
