| import Mathlib |
| import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives |
| import OAI.Geometry.PrescribedPotential.PatchCutoffs |
| import OAI.Geometry.PrescribedPotential.RealVolume |
| import OAI.Geometry.PrescribedPotential.RealPoisson |
| import OAI.Geometry.PrescribedPotential.RealPoissonEquivalence |
|
|
| namespace OAI |
|
|
| /-! Augmented Volume. -/ |
|
|
| section |
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| |
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|
| noncomputable section |
| open Set Filter Topology |
| open scoped ContDiff Classical |
| namespace ExponentialAugment |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| lemma derivative (V : E → F) (L : E →L[ℝ] F) (C : ℝ →L[ℝ] F) (e : E →L[ℝ] ℝ) |
| (hv : HasStrictFDerivAt V L 0) (h0 : V 0 = C 1) : |
| HasStrictFDerivAt (fun z : E × ℝ => (Real.exp (-z.2) • V z.1, e z.1)) |
| (((L ∘L ContinuousLinearMap.fst ℝ E ℝ) - (C ∘L ContinuousLinearMap.snd ℝ E ℝ)).prod |
| (e ∘L ContinuousLinearMap.fst ℝ E ℝ)) 0 := by |
| let π₁ := ContinuousLinearMap.fst ℝ E ℝ |
| let π₂ := ContinuousLinearMap.snd ℝ E ℝ |
| have hv' : HasStrictFDerivAt (fun z : E × ℝ => V z.1) (L ∘L π₁) 0 := |
| hv.comp 0 π₁.hasStrictFDerivAt |
| have he : HasStrictFDerivAt (fun z : E × ℝ => Real.exp (-z.2)) (-π₂) 0 := by |
| simpa [π₂] using ((-π₂).hasStrictFDerivAt (x := (0 : E × ℝ))).exp |
| have hh := he.smul hv' |
| have hn := (e ∘L π₁).hasStrictFDerivAt (x := (0 : E × ℝ)) |
| have hj := hh.prodMk hn |
| have hlin : (Real.exp (-(0 : E × ℝ).2) • (L ∘L π₁) + |
| (-π₂).smulRight (V (0 : E × ℝ).1)).prod (e ∘L π₁) = |
| ((L ∘L π₁) - (C ∘L π₂)).prod (e ∘L π₁) := by |
| apply ContinuousLinearMap.ext |
| intro z |
| apply Prod.ext |
| · change Real.exp (-0) • L z.1 + (-z.2) • V 0 = L z.1 - C z.2 |
| rw [neg_zero, Real.exp_zero, one_smul, h0, ← map_smul] |
| simp [sub_eq_add_neg] |
| · rfl |
| rw [hlin] at hj |
| exact hj |
| end ExponentialAugment |
|
|
| namespace GlobalElliptic |
| open Anticanonical SourceSmooth EllipticKernel SobolevChart |
| variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X] |
| {A : ComplexAtlas d X} {ι : Type*} [Fintype ι] |
| namespace GluingData |
| variable {g : KaehlerMetric A} (D : GluingData g ι) |
|
|
| local instance augmentedVolumeNormedAddCommGroup (s : ℝ) : |
| NormedAddCommGroup (D.localizers.RealSobolev s) := |
| (D.localizers.realCompletion s).normedAddCommGroup |
| local instance augmentedVolumeNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) := |
| (D.localizers.realCompletion s).normedSpace |
| local instance augmentedVolumeIsTopologicalAddGroup (s : ℝ) : |
| IsTopologicalAddGroup (D.localizers.RealSobolev s) := |
| Submodule.isTopologicalAddGroup _ |
| local instance augmentedVolumeContinuousSMul (s : ℝ) : |
| ContinuousSMul ℝ (D.localizers.RealSobolev s) := |
| SMulMemClass.continuousSMul _ |
|
|
| def augmentedVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) |
| (z : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) : |
| D.localizers.RealSobolev (k : ℝ) × ℝ := |
| (Real.exp (-z.2) • D.realVolume k hk z.1, D.realEvaluation ((k : ℝ)+2) x₀ z.1) |
|
|
| lemma augmentedVolume_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) : |
| D.augmentedVolume k hk x₀ 0 = (D.realConstants (k : ℝ) 1, 0) := by |
| simp [augmentedVolume, D.realVolume_zero] |
|
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| lemma augmentedVolume_hasStrictFDerivAt_zero (k : ℕ) |
| (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) : |
| HasStrictFDerivAt (D.augmentedVolume k hk x₀) (D.realAugmentedL k x₀) 0 := by |
| exact ExponentialAugment.derivative (D.realVolume k hk) (D.realLOrder k) |
| (D.realConstants (k : ℝ)) (D.realEvaluation ((k : ℝ)+2) x₀) |
| (D.realVolume_hasStrictFDerivAt_zero k hk) (D.realVolume_zero k hk) |
|
|
| end GluingData |
| end GlobalElliptic |
|
|
| end |
| end |
|
|
| end OAI |
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|
