| import Mathlib | |
| namespace OAI | |
| /-! Equicontinuity and finite-dimensionality of holomorphic sections on compact manifolds. -/ | |
| noncomputable section | |
| open Bundle Set Filter | |
| open scoped Manifold Bundle Topology BoundedContinuousFunction | |
| namespace CompactSections | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] | |
| [NormedAddCommGroup F] [NormedSpace ℂ F] | |
| theorem equicontinuousAt_of_ball_bound {ι : Type*} (f : ι → E → F) | |
| (x : E) {r C : ℝ} (hr : 0 < r) | |
| (hd : ∀ i, DifferentiableOn ℂ (f i) (Metric.ball x r)) | |
| (hb : ∀ i y, y ∈ Metric.ball x r → ‖f i y‖ ≤ C) : | |
| EquicontinuousAt f x := by | |
| apply Metric.equicontinuousAt_iff.mpr | |
| intro ε hε | |
| let D := 2 * max C 0 / r | |
| have hD : 0 ≤ D := div_nonneg (mul_nonneg (by norm_num) (le_max_right _ _)) hr.le | |
| refine ⟨min r (ε / (D + 1)), lt_min hr (div_pos hε (by positivity)), ?_⟩ | |
| intro y hy i | |
| have hyr : y ∈ Metric.ball x r := hy.trans_le (min_le_left _ _) | |
| have hmap : MapsTo (f i) (Metric.ball x r) (Metric.closedBall (f i x) (2 * max C 0)) := by | |
| intro z hz | |
| change dist (f i z) (f i x) ≤ _ | |
| calc | |
| dist (f i z) (f i x) ≤ ‖f i z‖ + ‖f i x‖ := dist_le_norm_add_norm _ _ | |
| _ ≤ C + C := add_le_add (hb i z hz) (hb i x (Metric.mem_ball_self hr)) | |
| _ ≤ 2 * max C 0 := by linarith [le_max_left C 0] | |
| have hSchwarz := Complex.dist_le_div_mul_dist_of_mapsTo_ball (hd i) hmap hyr | |
| rw [dist_comm] at hSchwarz | |
| apply hSchwarz.trans_lt | |
| have hyε : dist y x < ε / (D + 1) := hy.trans_le (min_le_right _ _) | |
| have hstrict : (D + 1) * dist y x < ε := by | |
| nlinarith [(lt_div_iff₀ (show 0 < D + 1 by positivity)).mp hyε] | |
| change D * dist y x < ε | |
| nlinarith [dist_nonneg (α := E) (x := y) (y := x)] | |
| theorem equicontinuousAt_clm_apply {X ι : Type*} [TopologicalSpace X] | |
| (f : ι → X → F) (A : X → F →L[ℂ] F) (x : X) | |
| (hf : EquicontinuousAt f x) (hA : ContinuousAt A x) | |
| {C : ℝ} (hb : ∀ i, ‖f i x‖ ≤ C) : | |
| EquicontinuousAt (fun i y => A y (f i y)) x := by | |
| apply Metric.equicontinuousAt_iff_right.mpr | |
| intro ε hε | |
| let K := ‖A x‖ + 1 | |
| let L := max C 0 + 1 | |
| have hK : 0 < K := by dsimp [K]; positivity | |
| have hL : 0 < L := by dsimp [L]; positivity | |
| have hsmall := (Metric.equicontinuousAt_iff_right.mp hf) | |
| (ε / 2 / K) (div_pos (half_pos hε) hK) | |
| have hopen : ∀ᶠ y in 𝓝 x, ‖A y - A x‖ < min 1 (ε / 2 / L) := by | |
| simpa only [dist_eq_norm] using | |
| (Metric.tendsto_nhds.mp hA _ (lt_min zero_lt_one (div_pos (half_pos hε) hL))) | |
| filter_upwards [hsmall, hopen] with y hy hAy | |
| intro i | |
| have hnorm : ‖A y‖ ≤ K := by | |
| have := norm_le_norm_add_norm_sub (A x) (A y) | |
| rw [norm_sub_rev] at this | |
| dsimp [K] | |
| linarith [lt_of_lt_of_le hAy (min_le_left _ _)] | |
| have h₁ : dist (A y (f i y)) (A y (f i x)) < ε / 2 := by | |
| calc | |
| dist (A y (f i y)) (A y (f i x)) ≤ ‖A y‖ * dist (f i y) (f i x) := | |
| (A y).dist_le_opNorm _ _ | |
| _ ≤ K * dist (f i x) (f i y) := by rw [dist_comm]; gcongr | |
| _ < ε / 2 := by | |
| have := (lt_div_iff₀ hK).mp (hy i) | |
| nlinarith | |
| have h₂ : dist (A y (f i x)) (A x (f i x)) < ε / 2 := by | |
| calc | |
| dist (A y (f i x)) (A x (f i x)) = ‖(A y - A x) (f i x)‖ := by | |
| rw [dist_eq_norm]; rfl | |
| _ ≤ ‖A y - A x‖ * ‖f i x‖ := (A y - A x).le_opNorm _ | |
| _ ≤ ‖A y - A x‖ * L := by | |
| gcongr | |
| exact (hb i).trans (by dsimp [L]; linarith [le_max_left C 0]) | |
| _ < ε / 2 := by | |
| have := (lt_div_iff₀ hL).mp (lt_of_lt_of_le hAy (min_le_right _ _)) | |
| exact this | |
| rw [dist_comm] | |
| exact (dist_triangle _ (A y (f i x)) _).trans_lt (by linarith) | |
| section ManifoldLocal | |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace E M] | |
| [IsManifold 𝓘(ℂ, E) (⊤ : WithTop ℕ∞) M] | |
| theorem equicontinuousAt_of_local_holomorphic_bound {ι : Type*} (f : ι → M → F) | |
| {U : Set M} (hU : IsOpen U) {x : M} (hx : x ∈ U) {C : ℝ} | |
| (hd : ∀ i, MDifferentiableOn 𝓘(ℂ, E) 𝓘(ℂ, F) (f i) U) | |
| (hb : ∀ i y, y ∈ U → ‖f i y‖ ≤ C) : | |
| EquicontinuousAt f x := by | |
| let e := extChartAt 𝓘(ℂ, E) x | |
| have het : IsOpen e.target := isOpen_extChartAt_target x | |
| have hex : e x ∈ e.target := mem_extChartAt_target x | |
| have hec : ContinuousAt e.symm (e x) := | |
| (contMDiffOn_extChartAt_symm (I := 𝓘(ℂ, E)) (n := (⊤ : WithTop ℕ∞)) x).continuousOn.continuousAt | |
| (het.mem_nhds hex) | |
| have hpre : e.symm ⁻¹' U ∈ 𝓝 (e x) := by | |
| apply hec.preimage_mem_nhds | |
| simpa only [e, extChartAt_to_inv] using hU.mem_nhds hx | |
| obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (inter_mem (het.mem_nhds hex) hpre) | |
| have heq : EquicontinuousAt (fun i => f i ∘ e.symm) (e x) := by | |
| refine equicontinuousAt_of_ball_bound (C := C) _ _ hr ?_ ?_ | |
| · intro i y hy | |
| have hyt := (hball hy).1 | |
| have hyU := (hball hy).2 | |
| have hemd : MDifferentiableAt 𝓘(ℂ, E) 𝓘(ℂ, E) e.symm y := | |
| ((contMDiffOn_extChartAt_symm (I := 𝓘(ℂ, E)) | |
| (n := (⊤ : WithTop ℕ∞)) x y hyt).contMDiffAt (het.mem_nhds hyt)).mdifferentiableAt (by simp) | |
| exact (mdifferentiableAt_iff_differentiableAt.mp | |
| (((hd i _ hyU).mdifferentiableAt (hU.mem_nhds hyU)).comp y hemd)).differentiableWithinAt | |
| · intro i y hy | |
| exact hb i _ (hball hy).2 | |
| apply Metric.equicontinuousAt_iff_right.mpr | |
| intro ε hε | |
| have hε' := (Metric.equicontinuousAt_iff_right.mp heq ε hε) | |
| have hecont : ContinuousAt e x := | |
| (contMDiffAt_extChartAt (I := 𝓘(ℂ, E)) (n := (⊤ : WithTop ℕ∞))).continuousAt | |
| filter_upwards [hecont hε', extChartAt_source_mem_nhds (I := 𝓘(ℂ, E)) x] with y hy hys | |
| intro i | |
| simpa only [Function.comp_apply, e.left_inv hys, e, extChartAt_to_inv] using hy i | |
| end ManifoldLocal | |
| section CompactCover | |
| variable [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] | |
| {M : Type*} [TopologicalSpace M] [T2Space M] | |
| [ChartedSpace E M] [IsManifold 𝓘(ℂ, E) (⊤ : WithTop ℕ∞) M] | |
| (V : M → Type*) [TopologicalSpace (TotalSpace F V)] | |
| [∀ x, TopologicalSpace (V x)] [∀ x, AddCommGroup (V x)] [∀ x, Module ℂ (V x)] | |
| [FiberBundle F V] [VectorBundle ℂ F V] | |
| [ContMDiffVectorBundle (⊤ : WithTop ℕ∞) F V 𝓘(ℂ, E)] | |
| {ι : Type*} [Fintype ι] | |
| (e : ι → Trivialization F (TotalSpace.proj : TotalSpace F V → M)) | |
| [∀ i, MemTrivializationAtlas (e i)] | |
| (K : ι → Set M) (hK : ∀ i, IsCompact (K i)) (hsub : ∀ i, K i ⊆ (e i).baseSet) | |
| abbrev HolSections := ContMDiffSection 𝓘(ℂ, E) F (⊤ : WithTop ℕ∞) V | |
| def sectionProfile : HolSections (E := E) (F := F) V →ₗ[ℂ] (∀ i, K i →ᵇ F) where | |
| toFun s i := by | |
| letI : CompactSpace (K i) := (isCompact_iff_compactSpace.mp (hK i)) | |
| exact BoundedContinuousFunction.mkOfCompact | |
| ⟨(fun x => (e i ⟨x.1, s x.1⟩).2), | |
| (((e i).contMDiffOn_section_baseSet_iff.mp s.contMDiff.contMDiffOn).continuousOn.mono | |
| (hsub i)).domRestrict⟩ | |
| map_add' s t := by | |
| funext i | |
| apply BoundedContinuousFunction.ext | |
| intro x | |
| exact ((e i).linearEquivAt ℂ x.1 (hsub i x.2)).map_add (s x.1) (t x.1) | |
| map_smul' c s := by | |
| funext i | |
| apply BoundedContinuousFunction.ext | |
| intro x | |
| exact ((e i).linearEquivAt ℂ x.1 (hsub i x.2)).map_smul c (s x.1) | |
| omit [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] [T2Space M] | |
| [IsManifold 𝓘(ℂ, E) ⊤ M] [Fintype ι] in | |
| theorem sectionProfile_injective (hcover : ∀ x, ∃ i, x ∈ interior (K i)) : | |
| Function.Injective (sectionProfile (E := E) V e K hK hsub) := by | |
| intro s t h | |
| ext x | |
| obtain ⟨i, hi⟩ := hcover x | |
| have hxK := interior_subset hi | |
| apply ((e i).linearEquivAt ℂ x (hsub i hxK)).injective | |
| exact congrArg (fun f : ∀ i, K i →ᵇ F => f i ⟨x, hxK⟩) h | |
| include hsub in | |
| omit [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] [T2Space M] [Fintype ι] in | |
| theorem sectionProfiles_equicontinuous {S : Type*} | |
| (s : S → HolSections (E := E) (F := F) V) | |
| (hcover : ∀ x, ∃ i, x ∈ interior (K i)) {C : ℝ} | |
| (hb : ∀ t i x, x ∈ K i → ‖(e i ⟨x, s t x⟩).2‖ ≤ C) (i : ι) : | |
| Equicontinuous (fun t (x : K i) => (e i ⟨x.1, s t x.1⟩).2) := by | |
| intro x | |
| obtain ⟨j, hj⟩ := hcover x.1 | |
| have hlocal : EquicontinuousAt (fun t y => (e j ⟨y, s t y⟩).2) x.1 := by | |
| refine equicontinuousAt_of_local_holomorphic_bound (E := E) (C := C) _ isOpen_interior hj ?_ ?_ | |
| · intro t | |
| exact (((e j).contMDiffOn_section_baseSet_iff.mp | |
| (s t).contMDiff.contMDiffOn).mdifferentiableOn (by simp)).mono | |
| (interior_subset.trans (hsub j)) | |
| · intro t y hy | |
| exact hb t j y (interior_subset hy) | |
| let A : M → F →L[ℂ] F := fun y => (e j).coordChangeL ℂ (e i) y | |
| have hxbase : x.1 ∈ (e j).baseSet ∩ (e i).baseSet := | |
| ⟨hsub j (interior_subset hj), hsub i x.2⟩ | |
| have hopen : IsOpen ((e j).baseSet ∩ (e i).baseSet) := | |
| (e j).open_baseSet.inter (e i).open_baseSet | |
| have hA : ContinuousAt A x.1 := | |
| (continuousOn_coordChange ℂ (e j) (e i)).continuousAt (hopen.mem_nhds hxbase) | |
| have heqc := equicontinuousAt_clm_apply _ A x.1 hlocal hA | |
| (fun t => hb t j x.1 (interior_subset hj)) | |
| have hid (t : S) (y : M) (hy : y ∈ (e j).baseSet ∩ (e i).baseSet) : | |
| A y ((e j ⟨y, s t y⟩).2) = (e i ⟨y, s t y⟩).2 := by | |
| dsimp [A] | |
| rw [(e j).coordChangeL_apply (e i) hy, (e j).symm_apply_apply_mk hy.1] | |
| apply Metric.equicontinuousAt_iff_right.mpr | |
| intro ε hε | |
| have hev := (Metric.equicontinuousAt_iff_right.mp heqc ε hε).and | |
| (hopen.mem_nhds hxbase) | |
| have hev' := (continuous_subtype_val.continuousAt (x := x)) hev | |
| filter_upwards [hev'] with y hy | |
| intro t | |
| simpa only [hid t x.1 hxbase, hid t y.1 hy.2] using hy.1 t | |
| include e hK hsub in | |
| omit [FiniteDimensional ℂ E] [T2Space M] in | |
| theorem finiteDimensional_of_compact_cover | |
| (hcover : ∀ x, ∃ i, x ∈ interior (K i)) : | |
| FiniteDimensional ℂ (HolSections (E := E) (F := F) V) := by | |
| classical | |
| let H := HolSections (E := E) (F := F) V | |
| let P := ∀ i, K i →ᵇ F | |
| let p : H →ₗ[ℂ] P := sectionProfile (E := E) V e K hK hsub | |
| have hp : Function.Injective p := sectionProfile_injective V e K hK hsub hcover | |
| let : NormedAddCommGroup H := NormedAddCommGroup.induced H P p hp | |
| let : NormedSpace ℂ H := NormedSpace.induced ℂ H P p | |
| have hiso : Isometry p := AddMonoidHomClass.isometry_of_norm p (fun _ => rfl) | |
| let B : Set H := Metric.closedBall 0 1 | |
| have hb (s : B) (i : ι) (x : M) (hx : x ∈ K i) : ‖(e i ⟨x, s.1 x⟩).2‖ ≤ 1 := by | |
| calc | |
| ‖(e i ⟨x, s.1 x⟩).2‖ = ‖p s.1 i ⟨x, hx⟩‖ := rfl | |
| _ ≤ ‖p s.1 i‖ := (p s.1 i).norm_coe_le_norm _ | |
| _ ≤ ‖p s.1‖ := norm_le_pi_norm _ i | |
| _ ≤ 1 := by | |
| change ‖s.1‖ ≤ 1 | |
| simpa only [B, Metric.mem_closedBall, dist_zero_right] using s.2 | |
| let A : ∀ i, Set (K i →ᵇ F) := fun i => (fun s : H => p s i) '' B | |
| have hcompact (i : ι) : IsCompact (closure (A i)) := by | |
| let : CompactSpace (K i) := isCompact_iff_compactSpace.mp (hK i) | |
| refine BoundedContinuousFunction.arzela_ascoli (Metric.closedBall 0 1) | |
| (isCompact_closedBall _ _) (A i) ?_ ?_ | |
| · intro f x hf | |
| obtain ⟨s, hs, rfl⟩ := hf | |
| rw [Metric.mem_closedBall, dist_zero_right] | |
| exact hb ⟨s, hs⟩ i x.1 x.2 | |
| · have heq := sectionProfiles_equicontinuous V e K hsub | |
| (fun s : B => s.1) hcover hb i | |
| intro x | |
| apply Metric.equicontinuousAt_iff_right.mpr | |
| intro ε hε | |
| filter_upwards [Metric.equicontinuousAt_iff_right.mp (heq x) ε hε] with y hy | |
| intro f | |
| obtain ⟨s, hs, hsf⟩ := f.2 | |
| change dist (f.1 x) (f.1 y) < ε | |
| rw [← hsf] | |
| exact hy ⟨s, hs⟩ | |
| have htotal : TotallyBounded (p '' B) := by | |
| apply (isCompact_pi_infinite hcompact).totallyBounded.subset | |
| rintro _ ⟨s, hs, rfl⟩ i | |
| exact subset_closure ⟨s, hs, rfl⟩ | |
| exact FiniteDimensional.of_totallyBounded_nhds_zero ℂ | |
| (Metric.closedBall_mem_nhds (0 : H) (by norm_num : (0 : ℝ) < 1)) | |
| ((totallyBounded_image_iff hiso.isUniformInducing).mp htotal) | |
| end CompactCover | |
| variable [FiniteDimensional ℂ E] [FiniteDimensional ℂ F] | |
| {M : Type*} [TopologicalSpace M] [T2Space M] [CompactSpace M] | |
| [ChartedSpace E M] [IsManifold 𝓘(ℂ, E) (⊤ : WithTop ℕ∞) M] | |
| (V : M → Type*) [TopologicalSpace (TotalSpace F V)] | |
| [∀ x, TopologicalSpace (V x)] [∀ x, AddCommGroup (V x)] [∀ x, Module ℂ (V x)] | |
| [FiberBundle F V] [VectorBundle ℂ F V] | |
| [ContMDiffVectorBundle (⊤ : WithTop ℕ∞) F V 𝓘(ℂ, E)] | |
| omit [FiniteDimensional ℂ E] in | |
| theorem finiteDimensional_holomorphic_sections : | |
| FiniteDimensional ℂ (ContMDiffSection 𝓘(ℂ, E) F (⊤ : WithTop ℕ∞) V) := by | |
| classical | |
| have hex (x : M) : ∃ K : Set M, IsCompact K ∧ x ∈ interior K ∧ | |
| K ⊆ (trivializationAt F V x).baseSet := | |
| exists_compact_subset (trivializationAt F V x).open_baseSet | |
| (mem_baseSet_trivializationAt F V x) | |
| choose K hK hxK hsub using hex | |
| obtain ⟨I, hI⟩ := isCompact_univ.elim_finite_subcover (fun x => interior (K x)) | |
| (fun _ => isOpen_interior) (fun x _ => mem_iUnion.mpr ⟨x, hxK x⟩) | |
| let e (i : I) := trivializationAt F V i.1 | |
| let (i : I) : MemTrivializationAtlas (e i) := inferInstanceAs | |
| (MemTrivializationAtlas (trivializationAt F V i.1)) | |
| apply finiteDimensional_of_compact_cover (E := E) V e (fun i => K i.1) | |
| (fun i => hK i.1) (fun i => hsub i.1) | |
| intro x | |
| obtain ⟨i, hi, hx⟩ := mem_iUnion₂.mp (hI (mem_univ x)) | |
| exact ⟨⟨i, hi⟩, hx⟩ | |
| end CompactSections | |
| end | |
| end OAI | |