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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
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
end
end
end OAI
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