| 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 |
|
|
