| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Contact.GraphCompact | |
| import OAI.Analysis.BiholderTransport.CostGeometry.BranchActiveGrowth | |
| import OAI.Analysis.BiholderTransport.Coordinates.CutCharacterization | |
| import OAI.Analysis.BiholderTransport.Regularity.LimitGeometry | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] | |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| lemma WeakMTW.isolated_graph_fiber_at_left_limit | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {T : ℝ} (hT : 0<T) (hT1 : T<1) | |
| (hinj : ∀ t ∈ Ioo (0:ℝ) T, Function.Injective (graphProjection (n := n) (cTransform v) t)) | |
| (z : subgradientGraph (n := n) (cTransform v)) : | |
| ∀ᶠ w in 𝓝 z, graphProjection (cTransform v) T w=graphProjection (cTransform v) T z → w=z := by | |
| have hglobal := graph_global_min_at_left_limit (continuous_cTransform hv) hT hinj | |
| have hmin (x : M) : ∀ p ∈ convexHull ℝ (activeLogs (n := n) v x), T • p ∈ minimizingVectors x := by | |
| intro p hp | |
| exact (hglobal ⟨⟨x,p⟩,active_hull_subset_normalSubdifferential hv x hp⟩).1 | |
| have hp := normalSubdifferential_subset_active_hull hv z.1.1 z.2 | |
| have hreg := hmtw.active_hull_nonconjugate_of_minimizing hT hT1 (hmin z.1.1) hp | |
| obtain ⟨G,hG,hagree⟩ := exists_smooth_cost_branch (tangentScale T z.1) hreg | |
| have hbranch := cost_branch_agrees_on_minimizing hG hagree | |
| have hbase := hmtw.active_hull_branch_strict_growth hv hp hT hT1 | |
| (active_hull_interior_below_minimizing hT (hmin z.1.1)) hG hagree | |
| obtain ⟨U,hU,hzU,hUinj⟩ := joint_exp_locally_injective_of_nonconjugate (tangentScale T z.1) hreg | |
| have hscale : Continuous (fun w : subgradientGraph (n := n) (cTransform v) => tangentScale T w.1) := | |
| contMDiff_tangentScale.continuous.comp (continuous_const.prodMk continuous_subtype_val) | |
| have hproj : Continuous (fun w : subgradientGraph (n := n) (cTransform v) => w.1.1) := | |
| (FiberBundle.continuous_proj _ _).comp continuous_subtype_val | |
| have hcostz : cost z.1.1 (graphProjection (cTransform v) T z)= | |
| G (z.1.1,graphProjection (cTransform v) T z) := | |
| hbranch.self_of_nhds (hglobal z).1 | |
| filter_upwards [hscale.continuousAt.eventually hbranch, | |
| hscale.continuousAt.preimage_mem_nhds (hU.mem_nhds hzU), | |
| hproj.continuousAt.eventually hbase] with w hwG hwU hwbase he | |
| have hcostw : cost w.1.1 (graphProjection (cTransform v) T w)= | |
| G (w.1.1,graphProjection (cTransform v) T w) := hwG (hglobal w).1 | |
| have hsame : w.1.1=z.1.1 := by | |
| by_contra hn | |
| have hlt := hwbase hn | |
| change cTransform v z.1.1+G (z.1.1,graphProjection (cTransform v) T z)/T < | |
| cTransform v w.1.1+G (w.1.1,graphProjection (cTransform v) T z)/T at hlt | |
| rw [he] at hcostw | |
| rw [←hcostz,←hcostw] at hlt | |
| have hle := (hglobal w).2 z.1.1 | |
| rw [he] at hle | |
| exact (not_lt_of_ge hle) hlt | |
| apply Subtype.ext | |
| apply tangentScale_injective hT.ne' | |
| exact hUinj hwU hzU (Prod.ext hsame he) | |
| lemma WeakMTW.injective_graphProjection_at_left_limit | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {T : ℝ} (hT : 0<T) (hT1 : T<1) | |
| (hinj : ∀ t ∈ Ioo (0:ℝ) T, Function.Injective (graphProjection (n := n) (cTransform v) t)) : | |
| Function.Injective (graphProjection (n := n) (cTransform v) T) := by | |
| let : CompactSpace (subgradientGraph (n := n) (cTransform v)) := | |
| isCompact_iff_compactSpace.mp (isCompact_subgradientGraph hv) | |
| let : LocallyConnectedSpace M := ChartedSpace.locallyConnectedSpace (Model n) M | |
| let : T2Space (TangentBundle 𝓘(ℝ,Model n) M) := bundle_totalSpace_t2 | |
| have hclosure : T ∈ closure {t | Function.Bijective (graphProjection (n := n) (cTransform v) t)} := by | |
| apply mem_closure_of_frequently_of_tendsto | |
| (f := fun t : ℝ => t) (b := 𝓝[<] T) | |
| · apply Filter.Eventually.frequently | |
| filter_upwards [(eventually_gt_nhds hT).filter_mono nhdsWithin_le_nhds, | |
| self_mem_nhdsWithin] with t ht htT | |
| exact ⟨hinj t ⟨ht,htT⟩,surjective_graphProjection (continuous_cTransform hv) ht⟩ | |
| · exact tendsto_id.mono_right nhdsWithin_le_nhds | |
| intro z w he | |
| have hconn := isPreconnected_fiber_of_mem_closure_bijective | |
| (graphProjection (n := n) (cTransform v)) (continuous_graphProjection_family _) hclosure | |
| (graphProjection (cTransform v) T z) | |
| have hs : {q : subgradientGraph (n := n) (cTransform v) | graphProjection (cTransform v) T q=graphProjection (cTransform v) T z}.Subsingleton := by | |
| apply fiber_subsingleton_of_preconnected_isolated hconn | |
| intro q hq | |
| obtain ⟨U,hUP,hU,hqU⟩ := mem_nhds_iff.mp (hmtw.isolated_graph_fiber_at_left_limit hv hT hT1 hinj q) | |
| refine ⟨U,hU,hqU,?_⟩ | |
| intro r hr | |
| exact hUP hr.1 (hr.2.trans hq.symm) | |
| exact hs rfl he.symm | |
| lemma WeakMTW.active_hull_precut_at_left_limit | |
| (hmtw : WeakMTW (n := n) (M := M)) {v : M → ℝ} (hv : Continuous v) | |
| {T : ℝ} (hT : 0<T) (hT1 : T<1) | |
| (hinj : ∀ t ∈ Ioo (0:ℝ) T, Function.Injective (graphProjection (n := n) (cTransform v) t)) : | |
| ∀ x : M, ∀ p ∈ convexHull ℝ (activeLogs (n := n) v x), T • p ∈ injectivityDomain x := by | |
| have hTI := hmtw.injective_graphProjection_at_left_limit hv hT hT1 hinj | |
| have hglobal := global_min_of_injective_graphProjection (continuous_cTransform hv) hT hTI | |
| have hmin (x : M) : ∀ p ∈ convexHull ℝ (activeLogs (n := n) v x), T • p ∈ minimizingVectors x := by | |
| intro p hp | |
| exact (hglobal ⟨⟨x,p⟩,active_hull_subset_normalSubdifferential hv x hp⟩).1 | |
| intro x p hp | |
| apply mem_injectivityDomain_of_nonconjugate_unique | |
| (hmtw.active_hull_nonconjugate_of_minimizing hT hT1 (hmin x) hp) | |
| intro q hq he | |
| let z : subgradientGraph (n := n) (cTransform v) := | |
| ⟨⟨x,p⟩,active_hull_subset_normalSubdifferential hv x hp⟩ | |
| exact unique_minimizing_direction_of_injective_graphProjection hT hTI z (hglobal z).2 hq he | |
| end WeakMTWTransport | |
| end | |
| end OAI | |