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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 | import Mathlib
import OAI.Geometry.PrescribedPotential.GlobalSobolev
import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
import OAI.Geometry.PrescribedRicci.TraceEntryBounds
import OAI.Geometry.PrescribedPotential.VolumePath
import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
import OAI.Geometry.PrescribedRicci.MongeAmpereL2
import OAI.Geometry.PrescribedRicci.TraceMaximum
namespace OAI
/-! Compact Path Bounds. -/
section
noncomputable section
open Matrix Set Filter Topology
open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
namespace Anticanonical.SourceSmooth.KaehlerMetric
variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
[ConnectedSpace X] {A : ComplexAtlas d X}
theorem volumePath_metric_bounds_on_compact (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
(hd : 2 ≤ d) (q : Fin A.count) {L : Set (Coordinates d)}
(hL : IsCompact L) (hLt : L ⊆ (A.chart q).target) : ∃ K : ℝ, 0 < K ∧
∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
(∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
∀ z ∈ L, ‖(g.deform φ hp).matrix q z‖ ≤ K ∧
‖((g.deform φ hp).matrix q z)⁻¹‖ ≤ K := by
have hd' : 0 < d := by omega
let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd'
obtain ⟨T,hT,htrace⟩ := g.volumePath_trace_bound line hd
obtain ⟨m,hm⟩ := hL.bddAbove_image
((g.backgroundSize_continuousOn (prescribedForcing g line) q).mono hLt)
let M := max 1 m
have hM : 1 ≤ M := le_max_left _ _
have hbg (z : Coordinates d) (hz : z ∈ L) :
g.BackgroundComponentBound (prescribedForcing g line) q z M :=
g.backgroundComponents_of_size _ q z ((hm (mem_image_of_mem _ hz)).trans (le_max_right _ _))
have hM0 : 0 ≤ M := by linarith
obtain ⟨D,hD,hdet⟩ := MongeAmpere.compact_matrix_bound
(show Continuous (fun H : Matrix (Fin d) (Fin d) ℂ => H.det.re) from
Complex.continuous_re.comp continuous_id.matrix_det) M
obtain ⟨Q,hQ,hmetric⟩ := MongeAmpere.metric_norm_bound_from_inverse (n:=Fin d) (M*T)
let E := Real.exp (2*‖(⟨(prescribedForcing g line).value,
(prescribedForcing g line).continuous⟩ : C(X,ℝ))‖)
have hE : 0 < E := Real.exp_pos _
refine ⟨1+M*T+E*D*Q,by positivity,?_⟩
intro φ hp hmean t b ht heq z hzL
have hz := hLt hzL
let x := (A.chart q).symm z
have hxs := (A.chart q).map_target hz
have hxz : A.chart q x = z := (A.chart q).right_inv hz
let H := (g.deform φ hp).matrix q z
have hH := (g.deform φ hp).positive q z hz
have hbi : ‖H⁻¹‖ ≤ M*T := by
apply MongeAmpere.matrix_norm_le_of_entries (mul_nonneg hM0 hT.le)
intro i j
have hi := MongeAmpere.entry_le_relative_trace H⁻¹ (g.matrix q z) hH.inv
(g.positive q _ hz) hM0
(fun i => (Complex.re_le_norm _).trans ((hbg z hzL).2.1 i i)) i j
have hs := htrace φ hp hmean t b ht heq x
change (g.deform φ hp).traceMetricValue g x ≤ T at hs
rw [(g.deform φ hp).traceMetricValue_local g q hxs,hxz] at hs
exact hi.trans (mul_le_mul_of_nonneg_left hs hM0)
have hdh : H.det.re ≤ E*D := by
have hv := g.density_mul_volume φ q hz
change (g.potentialDensity φ).value x*(g.matrix q z).det.re = H.det.re at hv
rw [← hv]
have hρ := (g.volumePath_density_bounds line ht hp heq x).2
change (g.potentialDensity φ).value x ≤ E at hρ
exact mul_le_mul hρ
(hdet _ (MongeAmpere.matrix_norm_le_of_entries hM0 (hbg z hzL).1))
(Complex.pos_iff.mp (g.positive q _ hz).det_pos).1.le hE.le
constructor
· have hb := (hmetric H hH hbi).trans (mul_le_mul_of_nonneg_right hdh hQ.le)
change ‖H‖ ≤ _
nlinarith [mul_nonneg hM0 hT.le]
· change ‖H⁻¹‖ ≤ _
nlinarith [mul_nonneg (mul_nonneg hE.le hD.le) hQ.le]
end Anticanonical.SourceSmooth.KaehlerMetric
end
end
end OAI
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