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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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | import Mathlib
import OAI.Geometry.PrescribedPotential.AugmentedVolume
import OAI.Geometry.PrescribedPotential.RealNonlinearDerivative
namespace OAI
/-! Closed Range Smooth. -/
section
noncomputable section
open Set Filter Topology
open scoped ContDiff Classical
namespace ClosedRangeSmooth
universe u
variable {E F G : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
lemma isometry_derivative_iff (e : F →ₗᵢ[ℝ] G) (f : E → F) (L : E →L[ℝ] F) (x : E) :
HasFDerivAt f L x ↔ HasFDerivAt (fun y => e (f y)) (e.toContinuousLinearMap ∘L L) x := by
simp only [hasFDerivAt_iff_isLittleO]
conv_lhs => rw [← Asymptotics.isLittleO_norm_left]
conv_rhs => rw [← Asymptotics.isLittleO_norm_left]
simp only [ContinuousLinearMap.comp_apply, LinearIsometry.coe_toContinuousLinearMap,
← map_sub, e.norm_map]
lemma lift_derivative [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
{x : E} {J : E →L[ℝ] G} (hJ : HasFDerivAt (fun y => e (f y)) J x) :
∃ L : E →L[ℝ] F, HasFDerivAt f L x ∧ e.toContinuousLinearMap ∘L L = J := by
let S := e.range
have hS : IsClosed (S : Set G) := e.isometry.isClosedEmbedding.isClosed_range
let r : E → S := fun y => e.equivRange (f y)
have hmem : ∀ v, J v ∈ S :=
ClosedDerivative.derivative_mem S hS r hJ
let L := e.equivRange.symm.toContinuousLinearEquiv.toContinuousLinearMap ∘L J.codRestrict S hmem
have hcomp : e.toContinuousLinearMap ∘L L = J := by
ext v
change e (e.equivRange.symm ⟨J v,hmem v⟩) = J v
exact congrArg Subtype.val (e.equivRange.apply_symm_apply ⟨J v,hmem v⟩)
refine ⟨L,(isometry_derivative_iff e f L x).mpr ?_,hcomp⟩
rwa [hcomp]
lemma contDiff_of_isometry (n : ℕ) [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
(hf : ContDiff ℝ n (fun y => e (f y))) : ContDiff ℝ n f := by
induction n generalizing F G with
| zero =>
rw [Nat.cast_zero, contDiff_zero] at hf ⊢
exact e.isEmbedding.continuous_iff.mpr hf
| succ n ih =>
rw [Nat.cast_succ, contDiff_succ_iff_hasFDerivAt] at hf ⊢
obtain ⟨J,hJ,hd⟩ := hf
choose L hL he using fun x => lift_derivative e f (hd x)
refine ⟨L,?_,hL⟩
let ep : (E →L[ℝ] F) →ₗᵢ[ℝ] (E →L[ℝ] G) := e.postcomp
apply ih ep L
convert hJ using 1
funext x
exact he x
lemma contDiff_infty_of_isometry [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
(hf : ContDiff ℝ ∞ (fun y => e (f y))) : ContDiff ℝ ∞ f := by
apply contDiff_infty.mpr
intro n
exact contDiff_of_isometry n e f (contDiff_infty.mp hf n)
end ClosedRangeSmooth
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 closedRangeNormedGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
(D.localizers.realCompletion s).normedAddCommGroup
local instance closedRangeNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
(D.localizers.realCompletion s).normedSpace
local instance closedRangeTopologicalGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
Submodule.isTopologicalAddGroup _
local instance closedRangeContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
SMulMemClass.continuousSMul _
lemma realVolume_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
ContDiff ℝ ∞ (D.realVolume k hk) :=
ClosedRangeSmooth.contDiff_infty_of_isometry
(D.localizers.realCompletion (k : ℝ)).subtypeₗᵢ _ (D.realVolume_ambient_contDiff k hk)
lemma augmentedVolume_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
ContDiff ℝ ∞ (D.augmentedVolume k hk x₀) := by
apply ContDiff.prodMk
· exact (contDiff_snd.neg.exp).smul ((D.realVolume_contDiff k hk).comp contDiff_fst)
· exact (D.realEvaluation ((k : ℝ)+2) x₀).contDiff.comp contDiff_fst
end GluingData
end GlobalElliptic
end
end
end OAI
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