| import Mathlib |
| import OAI.Geometry.PrescribedRicci.ChernLuAlgebra |
| import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds |
| import OAI.Geometry.PrescribedRicci.VolumePathRicci |
| import OAI.Geometry.PrescribedPotential.VolumePath |
|
|
| namespace OAI |
|
|
| /-! Compact Path Ricci. -/ |
|
|
| section |
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| |
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|
| 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] {A : ComplexAtlas d X} |
|
|
| lemma volumePath_holDerivative_congr {f h : Coordinates d → Matrix (Fin d) (Fin d) ℂ} |
| {z : Coordinates d} (he : f =ᶠ[nhds z] h) (a : Fin d) : |
| holDerivative f z a = holDerivative h z a := by unfold holDerivative; rw [he.fderiv_eq] |
| |
| lemma holDerivative_sub_smul {f h : Coordinates d → Matrix (Fin d) (Fin d) ℂ} |
| {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (hh : DifferentiableAt ℝ h z) |
| (t : ℂ) (a : Fin d) : |
| holDerivative (fun y => f y-t • h y) z a = holDerivative f z a-t • holDerivative h z a := by |
| unfold holDerivative |
| have he : fderiv ℝ (fun y => f y-t • h y) z = fderiv ℝ f z-t • fderiv ℝ h z := by |
| convert! (hf.hasFDerivAt.sub (hh.hasFDerivAt.const_smul t)).fderiv using 1 |
| rw [he] |
| simp only [_root_.sub_apply,_root_.smul_apply,smul_sub, |
| smul_smul] |
| module |
|
|
| lemma volumePath_ricci_hol (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A) |
| {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ) |
| (heq : ∀ x, (g.logRatio (g.deform φ hp)).value x = |
| t*(prescribedForcing g line).value x+b) |
| (q : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) : |
| holDerivative ((g.deform φ hp).curvatureRicci q) z a = |
| holDerivative (g.curvatureRicci q) z a- |
| (t:ℂ) • holDerivative ((prescribedForcing g line).hessian q) z a := by |
| have he := Filter.eventually_of_mem ((A.chart q).open_target.mem_nhds hz) |
| (fun y hy => g.volumePath_ricci line hp heq q hy) |
| rw [volumePath_holDerivative_congr he,holDerivative_sub_smul |
| ((g.curvatureRicci_smooth q hz).differentiableAt (by simp)) |
| ((((prescribedForcing g line).hessian_smooth q).contDiffAt |
| ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp))] |
|
|
| lemma volumePath_ricci_bounds_on_compact (g : KaehlerMetric A) |
| (line : SemipositiveAnticanonicalMetric A) (q : Fin A.count) {L : Set (Coordinates d)} |
| (hL : IsCompact L) (hLt : L ⊆ (A.chart q).target) : |
| ∃ R : ℝ, 0 < R ∧ ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) |
| (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).curvatureRicci q z‖ ≤ R ∧ |
| ∀ a, ‖holDerivative ((g.deform φ hp).curvatureRicci q) z a‖ ≤ R := by |
| let F := prescribedForcing g line |
| let size : Coordinates d → ℝ := fun z => ‖g.curvatureRicci q z‖+‖F.hessian q z‖+ |
| ‖holDerivative (g.curvatureRicci q) z‖+‖holDerivative (F.hessian q) z‖ |
| have hs : ContinuousOn size L := by |
| intro z hz |
| have hR := g.curvatureRicci_smooth q (hLt hz) |
| have hF := (F.hessian_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds (hLt hz)) |
| have hdR : ContDiffAt ℝ ∞ (fun y => holDerivative (g.curvatureRicci q) y) z := |
| contDiffAt_pi.mpr (fun a => contDiffAt_holDerivative hR a) |
| have hdF : ContDiffAt ℝ ∞ (fun y => holDerivative (F.hessian q) y) z := |
| contDiffAt_pi.mpr (fun a => contDiffAt_holDerivative hF a) |
| exact (((hR.continuousAt.norm.add hF.continuousAt.norm).add hdR.continuousAt.norm).add |
| hdF.continuousAt.norm).continuousWithinAt |
| obtain ⟨r,hr⟩ := hL.bddAbove_image hs |
| refine ⟨1+max 0 r,by positivity,?_⟩ |
| intro φ hp t b ht heq z hz |
| have htr : ‖(t:ℂ)‖ ≤ 1 := by simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] using ht.2 |
| have hb : size z ≤ 1+max 0 r := (hr (mem_image_of_mem _ hz)).trans (by linarith [le_max_right (0:ℝ) r]) |
| constructor |
| · rw [g.volumePath_ricci line hp heq q (hLt hz)] |
| apply (norm_sub_le _ _).trans |
| rw [norm_smul] |
| have hn := mul_le_mul_of_nonneg_right htr (norm_nonneg (F.hessian q z)) |
| change ‖g.curvatureRicci q z‖+‖(t:ℂ)‖*‖F.hessian q z‖ ≤ _ |
| dsimp [size] at hb |
| linarith [norm_nonneg (holDerivative (g.curvatureRicci q) z),norm_nonneg (holDerivative (F.hessian q) z)] |
| · intro a |
| rw [g.volumePath_ricci_hol line hp heq q (hLt hz) a] |
| apply (norm_sub_le _ _).trans |
| rw [norm_smul] |
| have hn := mul_le_mul_of_nonneg_right htr (norm_nonneg (holDerivative (F.hessian q) z a)) |
| have h1 := norm_le_pi_norm (holDerivative (g.curvatureRicci q) z) a |
| have h2 := norm_le_pi_norm (holDerivative (F.hessian q) z) a |
| change ‖holDerivative (g.curvatureRicci q) z a‖+‖(t:ℂ)‖*‖holDerivative (F.hessian q) z a‖ ≤ _ |
| dsimp [size] at hb |
| linarith [norm_nonneg (g.curvatureRicci q z),norm_nonneg (F.hessian q z)] |
|
|
| end Anticanonical.SourceSmooth.KaehlerMetric |
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|
| end |
| end |
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| end OAI |
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