port(exhaustion-cutoff): 穷竭函数 + 局部光滑谓词 + 单位分解⇒仿紧

OpenGALib/Manifold.lean

@@ -1,6 +1,7 @@
 import OpenGALib.Manifold.Charts.ChartedSpaceCore
 import OpenGALib.Manifold.Charts.CoordinateBall
 import OpenGALib.Manifold.Charts.PrecompactBasis
+import OpenGALib.Manifold.Cutoff.Exhaustion
 
 /-!
 # Manifold

OpenGALib/Manifold/Cutoff/Exhaustion.lean

@@ -0,0 +1,266 @@
+import Mathlib.Geometry.Manifold.ContMDiff.Basic
+import Mathlib.Geometry.Manifold.ContMDiffMap
+import Mathlib.Geometry.Manifold.PartitionOfUnity
+import Mathlib.Topology.PartitionOfUnity
+import Mathlib.Topology.Maps.Proper.CompactlyGenerated
+import Mathlib.Topology.Order.Compact
+import Mathlib.Tactic.MkIffOfInductiveProp
+
+/-!
+# Exhaustion functions and local smoothness
+
+Cutoff- and exhaustion-style machinery for non-compact manifolds, layered by
+how much structure each piece needs so downstream code imports only the
+generality it uses:
+
+1. **Topological base** — `IsExhaustionFunction` is stated on any topological
+   space: a continuous real function with compact closed sublevel sets. Its
+   consequences (`tendsto_atTop`, `isProperMap`) are purely topological and are
+   the reusable interface for proving properness and cocompact decay.
+2. **Manifold-local smoothness** — `IsSmoothOn` packages local smooth
+   extendability of a map on a subset of a charted space, with an unbundled
+   restatement for downstream use.
+3. **Existence on manifolds** — `exists_positive_smooth_exhaustion_function`
+   builds a positive smooth exhaustion function on any Hausdorff
+   `σ`-compact smooth manifold via partition-of-unity gluing; this is the
+   payload that supports analysis on non-compact manifolds (L²/Bochner/GMT).
+4. **Paracompactness criterion** — `paracompactSpace_of_hasSubordinatePartitionOfUnity`
+   recovers paracompactness from the availability of subordinate partitions of
+   unity, closing the loop with the existence layer.
+
+## Main definitions
+
+* `Manifold.IsSmoothOn` — a map on a subset is smooth when it locally extends to a smooth map.
+* `Manifold.IsExhaustionFunction` — continuous function with compact closed sublevel sets.
+
+## Main results
+
+* `Manifold.isSmoothOn_iff_exists_local_extension` — bundled/unbundled equivalence of `IsSmoothOn`.
+* `Manifold.IsExhaustionFunction.tendsto_atTop` — an exhaustion function tends to `+∞` off compact sets.
+* `Manifold.IsExhaustionFunction.isProperMap` — an exhaustion function is a proper map.
+* `Manifold.Continuous.isExhaustionFunction` — any continuous function on a compact space is an exhaustion function.
+* `Manifold.exists_positive_smooth_exhaustion_function` — every Hausdorff `σ`-compact smooth manifold admits a positive smooth exhaustion function.
+* `Manifold.paracompactSpace_of_hasSubordinatePartitionOfUnity` — subordinate partitions of unity for all open covers imply paracompactness.
+
+Provenance: SmoothManifoldsLee a5f308c — Definition_2_11_extra_2, Definition_2_11_extra_3, Proposition_2_28, Problem_2_13.
+-/
+
+open scoped Manifold ContDiff
+open TopologicalSpace Filter Set
+
+universe uK uE uH uM uE2 uH2 uN u
+
+namespace Manifold
+
+/-! ## Local smoothness on a subset -/
+
+section IsSmoothOn
+
+variable {𝕜 : Type uK} [NontriviallyNormedField 𝕜]
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {H : Type uH} [TopologicalSpace H]
+variable {M : Type uM} [TopologicalSpace M] [ChartedSpace H M]
+variable {E' : Type uE2} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
+variable {H' : Type uH2} [TopologicalSpace H']
+variable {N : Type uN} [TopologicalSpace N] [ChartedSpace H' N]
+variable {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'}
+
+/-- **Math.** A map defined on a subset `A` of a smooth manifold is *smooth* if
+every point of `A` has an open neighborhood on which the map extends to a smooth
+map. -/
+def IsSmoothOn {A : Set M} (f : A → N) (I : ModelWithCorners 𝕜 E H)
+    (I' : ModelWithCorners 𝕜 E' H') : Prop :=
+  ∀ p : A,
+    ∃ U : Opens M,
+      (p : M) ∈ U ∧
+        ∃ Fext : C^∞⟮I, U; I', N⟯,
+          ∀ q : A, (hq : (q : M) ∈ U) → Fext ⟨q, hq⟩ = f q
+
+/-- **Math.** `IsSmoothOn` is equivalent to the unbundled local-extension
+formulation: each point has an open set on which a globally-defined `ContMDiffOn`
+map agrees with `f`. -/
+theorem isSmoothOn_iff_exists_local_extension {A : Set M} {f : A → N} :
+    IsSmoothOn f I I' ↔
+      ∀ p : A,
+        ∃ U : Set M,
+          IsOpen U ∧
+            (p : M) ∈ U ∧
+              ∃ Fext : M → N,
+                ContMDiffOn I I' (∞ : ℕ∞ω) Fext U ∧
+                  ∀ q : A, (q : M) ∈ U → Fext q = f q := by
+  classical
+  constructor
+  · intro hf p
+    rcases hf p with ⟨U, hpU, Fext, hFext⟩
+    refine ⟨U, U.isOpen, hpU, ?_⟩
+    let G : M → N := fun x ↦ if hx : x ∈ U then Fext ⟨x, hx⟩ else Fext ⟨p, hpU⟩
+    refine ⟨G, ?_, ?_⟩
+    · intro x hx
+      have hG_subtype : ContMDiffAt I I' (∞ : ℕ∞ω) (fun y : U ↦ G y) ⟨x, hx⟩ := by
+        have hEq : (fun y : U ↦ G y) = Fext := by
+          funext y
+          change G y = Fext y
+          simp [G]
+        rw [hEq]
+        exact Fext.contMDiff ⟨x, hx⟩
+      exact (contMDiffAt_subtype_iff.1 hG_subtype).contMDiffWithinAt
+    · intro q hq
+      unfold G
+      split_ifs with h
+      · simpa using hFext q h
+      · exact (h hq).elim
+  · intro hf p
+    rcases hf p with ⟨U, hU, hpU, Fext, hFext, hEq⟩
+    let Uo : Opens M := ⟨U, hU⟩
+    refine ⟨Uo, hpU, ?_⟩
+    refine ⟨⟨fun x : Uo ↦ Fext x, ?_⟩, ?_⟩
+    · intro x
+      exact contMDiffAt_subtype_iff.2 <|
+        (hFext x x.property).contMDiffAt <|
+          hU.mem_nhds x.property
+    · intro q hq
+      exact hEq q hq
+
+end IsSmoothOn
+
+/-! ## Exhaustion functions -/
+
+section Exhaustion
+
+variable {M : Type u} [TopologicalSpace M]
+
+/-- **Math.** An *exhaustion function* on `M` is a continuous real-valued
+function whose closed sublevel set `f ⁻¹' Iic c` is compact for every `c : ℝ`. -/
+@[mk_iff isExhaustionFunction_iff]
+class IsExhaustionFunction (f : M → ℝ) : Prop where
+  continuous : Continuous f
+  isCompact_sublevelSet (c : ℝ) : IsCompact (f ⁻¹' Iic c)
+
+/-- **Math.** An exhaustion function tends to `+∞` away from compact sets. -/
+theorem IsExhaustionFunction.tendsto_atTop {f : M → ℝ} (hf : IsExhaustionFunction f) :
+    Tendsto f (cocompact M) atTop := by
+  rw [Filter.atTop_basis_Ioi.tendsto_right_iff]
+  intro c _
+  change f ⁻¹' Ioi c ∈ cocompact M
+  convert (hf.isCompact_sublevelSet c).compl_mem_cocompact using 1
+  ext x
+  simp [not_le]
+
+/-- **Math.** Every exhaustion function is a proper map. -/
+theorem IsExhaustionFunction.isProperMap {f : M → ℝ} (hf : IsExhaustionFunction f) :
+    IsProperMap f :=
+  isProperMap_iff_tendsto_cocompact.2 ⟨hf.continuous, hf.tendsto_atTop.trans atTop_le_cocompact⟩
+
+/-- **Math.** An exhaustion function canonically yields a proper map. -/
+instance instIsProperMapOfIsExhaustionFunction {f : M → ℝ} [hf : IsExhaustionFunction f] :
+    IsProperMap f :=
+  hf.isProperMap
+
+/-- **Math.** An exhaustion function canonically yields continuity of its
+underlying function. -/
+instance instContinuousOfIsExhaustionFunction {f : M → ℝ} [hf : IsExhaustionFunction f] :
+    Continuous f :=
+  hf.continuous
+
+/-- **Math.** Any continuous real-valued function on a compact space is an
+exhaustion function. -/
+theorem Continuous.isExhaustionFunction {M : Type u} [TopologicalSpace M] [CompactSpace M]
+    {f : M → ℝ} (hf : Continuous f) : IsExhaustionFunction f := by
+  refine ⟨hf, fun c ↦ ?_⟩
+  exact isCompact_univ.of_isClosed_subset (isClosed_Iic.preimage hf) (Set.subset_univ _)
+
+end Exhaustion
+
+/-! ## Positive smooth exhaustion functions on a manifold -/
+
+section SmoothExhaustion
+
+/-- **Math.** Every smooth manifold (with or without boundary) that is Hausdorff
+and $\sigma$-compact admits a smooth real-valued function that is everywhere
+positive and has compact closed sublevel sets. -/
+theorem exists_positive_smooth_exhaustion_function
+    {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+    {H : Type uH} [TopologicalSpace H]
+    {M : Type uM} [TopologicalSpace M] [ChartedSpace H M]
+    (I : ModelWithCorners ℝ E H) [IsManifold I (∞ : ℕ∞ω) M] [T2Space M] [SigmaCompactSpace M] :
+    ∃ f : C^∞⟮I, M; ℝ⟯, (∀ x : M, 0 < f x) ∧ IsExhaustionFunction (f : M → ℝ) := by
+  haveI : LocallyCompactSpace H := I.locallyCompactSpace
+  haveI : LocallyCompactSpace M := ChartedSpace.locallyCompactSpace H M
+  let K : CompactExhaustion M := default
+  let t : M → Set ℝ := fun x ↦ Set.Ioi (K.find x : ℝ)
+  -- The target set at each point is the open ray above the compact-exhaustion rank.
+  have ht : ∀ x, Convex ℝ (t x) := by
+    intro x
+    simp only [t]
+    exact convex_Ioi (K.find x : ℝ)
+  -- Near `x`, the constant `(K.find x : ℝ) + 2` stays above all nearby target thresholds.
+  have hloc : ∀ x : M, ∃ c : ℝ, ∀ᶠ y in nhds x, c ∈ t y := by
+    intro x
+    refine ⟨(K.find x : ℝ) + 2, ?_⟩
+    have hK : K (K.find x + 1) ∈ nhds x := by
+      exact mem_interior_iff_mem_nhds.mp (K.subset_interior_succ _ (K.mem_find x))
+    filter_upwards [hK] with y hy
+    simp only [t, Set.mem_Ioi]
+    have hy_find : K.find y ≤ K.find x + 1 := (K.mem_iff_find_le).1 hy
+    exact_mod_cast lt_of_le_of_lt hy_find (Nat.lt_succ_self (K.find x + 1))
+  -- A smooth partition-of-unity gluing theorem produces a smooth function in all targets.
+  obtain ⟨f, hf⟩ : ∃ f : C^∞⟮I, M; ℝ⟯, ∀ x : M, f x ∈ t x := by
+    simpa only [t] using exists_contMDiffMap_forall_mem_convex_of_local_const I ht hloc
+  refine ⟨f, ?_⟩
+  constructor
+  · -- Positivity follows because the compact-exhaustion rank is nonnegative everywhere.
+    intro x
+    have h_find_nonneg : 0 ≤ (K.find x : ℝ) := by
+      exact_mod_cast Nat.zero_le (K.find x)
+    have h_find_lt : (K.find x : ℝ) < f x := by
+      simpa [t, Set.mem_Ioi] using hf x
+    exact lt_of_le_of_lt h_find_nonneg h_find_lt
+  · -- Each closed sublevel set is contained in a compact stage of the compact exhaustion.
+    refine ⟨f.contMDiff.continuous, ?_⟩
+    intro c
+    refine
+      (K.isCompact ⌈c⌉₊).of_isClosed_subset (isClosed_Iic.preimage f.contMDiff.continuous) ?_
+    intro x hx
+    have h_find_lt : (K.find x : ℝ) < f x := by
+      simpa [t, Set.mem_Ioi] using hf x
+    have h_find_lt_c : (K.find x : ℝ) < c := h_find_lt.trans_le hx
+    apply (K.mem_iff_find_le).2
+    exact (Nat.lt_ceil.2 h_find_lt_c).le
+
+end SmoothExhaustion
+
+/-! ## Paracompactness from subordinate partitions of unity -/
+
+section Paracompact
+
+/-- **Math.** If every indexed open cover of `M` admits a partition of unity
+subordinate to that cover, then `M` is paracompact. -/
+theorem paracompactSpace_of_hasSubordinatePartitionOfUnity
+    {M : Type u} [TopologicalSpace M]
+    (hM : ∀ {ι : Type u} (U : ι → Set M), (∀ i, IsOpen (U i)) → (⋃ i, U i = Set.univ) →
+      ∃ f : PartitionOfUnity ι M Set.univ, f.IsSubordinate U) :
+    ParacompactSpace M := by
+  refine ⟨fun ι U hU_open hU_cover ↦ ?_⟩
+  obtain ⟨f, hf⟩ := hM U hU_open hU_cover
+  let V : ι → Set M := fun i ↦ (f i : M → ℝ) ⁻¹' Set.Ioi 0
+  refine ⟨ι, V, ?_, ?_, ?_, ?_⟩
+  · intro i
+    exact isOpen_Ioi.preimage (f i).continuous
+  · ext x
+    constructor
+    · intro _
+      simp
+    · intro _
+      rcases f.exists_pos (by simp : x ∈ Set.univ) with ⟨i, hi⟩
+      exact Set.mem_iUnion.2 ⟨i, by simpa [V] using hi⟩
+  · refine f.locallyFinite.subset fun i x hx ↦ ?_
+    change 0 < (f i : M → ℝ) x at hx
+    simpa [Function.mem_support] using (ne_of_gt hx)
+  · intro i
+    refine ⟨i, fun x hx ↦ hf i <| subset_tsupport (f i : M → ℝ) ?_⟩
+    change 0 < (f i : M → ℝ) x at hx
+    simpa [Function.mem_support] using (ne_of_gt hx)
+
+end Paracompact
+
+end Manifold