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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import Mathlib
import OAI.Geometry.PrescribedPotential.NormalizedPathCompactness
import OAI.Geometry.PrescribedPotential.SmoothPathOpenness
namespace OAI
/-! Bounded Path Continuation. -/
section
noncomputable section
open Set Filter Topology
open scoped ContDiff Classical
namespace GlobalElliptic
open Anticanonical SourceSmooth EllipticKernel SobolevChart
variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
[ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
namespace GluingData
variable {g : KaehlerMetric A} (D : GluingData g ι)
def PathSobolevBounds (h : SemipositiveAnticanonicalMetric A) (x₀ : X) : Prop :=
∀ k : ℕ, ∃ C : ℝ, ∀ (t b : ℝ) (φ : SmoothRealFunction A),
t ∈ Icc 0 1 → φ.value x₀ = 0 → SolvesVolumePath g h t φ b →
‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (Smooth.ofReal φ)‖ ≤ C
omit [ConnectedSpace X] in
lemma smoothVolumePathTimes_relatively_closed (h : SemipositiveAnticanonicalMetric A)
(x₀ : X) (hbound : D.PathSobolevBounds h x₀) :
IsClosed ((Subtype.val : Icc (0:ℝ) 1 → ℝ) ⁻¹' SmoothVolumePathTimes g h x₀) := by
apply IsSeqClosed.isClosed
intro t tLim ht htLim
choose φ b hn hs using ht
have hB : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (RealSmooth.ofReal (φ n)).val‖ ≤ C := by
intro k
obtain ⟨C,hC⟩ := hbound k
exact ⟨C,fun n => hC (t n).val (b n) (φ n) (t n).property (hn n) (hs n)⟩
have ht' : Tendsto (fun n => (t n).val) atTop (𝓝 tLim.val) :=
(continuous_subtype_val.tendsto tLim).comp htLim
obtain ⟨fLim,bLim,r,hr,hb,hf,hnLim,hsLim⟩ := D.normalized_path_sequence_compact h x₀
(fun n => (t n).val) b tLim.val ht' (fun n => (t n).property)
(fun n => RealSmooth.ofReal (φ n)) (by simpa only [RealSmooth.source_ofReal] using hn)
(by simpa only [RealSmooth.source_ofReal] using hs) hB
exact ⟨fLim.source,bLim,hnLim,hsLim⟩
theorem smoothVolumePathTimes_full_of_bounds (h : SemipositiveAnticanonicalMetric A)
(x₀ : X) (hbound : D.PathSobolevBounds h x₀) :
Icc 0 1 ⊆ SmoothVolumePathTimes g h x₀ := by
let S : Set (Icc (0:ℝ) 1) := Subtype.val ⁻¹' SmoothVolumePathTimes g h x₀
have hc : IsClopen S := ⟨D.smoothVolumePathTimes_relatively_closed h x₀ hbound,
(smoothVolumePathTimes_isOpen g h x₀).preimage continuous_subtype_val⟩
have hn : S.Nonempty := ⟨⟨0,by constructor <;> norm_num⟩,
smoothVolumePathTimes_zero g h x₀⟩
have he := hc.eq_univ hn
intro t ht
have hh : (⟨t,ht⟩ : Icc (0:ℝ) 1) ∈ S := by rw [he]; trivial
exact hh
end GluingData
end GlobalElliptic
namespace Anticanonical.SourceSmooth
open GlobalElliptic
variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
[ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
theorem prescribedPotential_of_path_bounds (P : A.ProjectiveEmbedding)
(h : SemipositiveAnticanonicalMetric A) (D : GluingData (projectiveBackground P) ι)
(x₀ : X) (hbound : D.PathSobolevBounds h x₀) : HasPrescribedVolumePotential P h := by
obtain ⟨φ,b,_,hs⟩ := D.smoothVolumePathTimes_full_of_bounds h x₀ hbound
(by constructor <;> norm_num : (1:ℝ) ∈ Icc 0 1)
exact volumePath_one P h φ b hs
end Anticanonical.SourceSmooth
end
end
end OAI
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