| import Mathlib |
| import OAI.Geometry.PrescribedRicci.CalabiCutoffDifferential |
| import OAI.Geometry.PrescribedRicci.CalabiGradient |
| import OAI.Geometry.PrescribedRicci.CalabiLower |
| import OAI.Geometry.PrescribedRicci.CalabiTraceBound |
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| namespace OAI |
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| /-! Calabi Local Maximum. -/ |
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| section |
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| noncomputable section |
| open Matrix Set Filter Topology |
| open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise |
| namespace Anticanonical.SourceSmooth.KaehlerMetric |
| open MongeAmpere |
| variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X} |
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| lemma flatTrace_nonneg (g : KaehlerMetric A) (q : Fin A.count) |
| {z : Coordinates d} (hz : z ∈ (A.chart q).target) : 0 ≤ g.flatTrace q z := |
| (Complex.nonneg_iff.mp (g.positive q z hz).posSemidef.trace_nonneg).1 |
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| lemma compact_cutoff_derivative_bound {L : Set (Coordinates d)} (hL : IsCompact L) |
| (χ : Coordinates d → ℝ) (hχ : ContDiff ℝ ∞ χ) : |
| ∃ B : ℝ, 0 < B ∧ ∀ z ∈ L, |χ z| ≤ B ∧ ‖holRealDeriv χ z‖ ≤ B ∧ |
| ‖PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z‖ ≤ B := by |
| let size : Coordinates d → ℝ := fun z => |χ z|+‖holRealDeriv χ z‖+ |
| ‖PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z‖ |
| have hs : Continuous size := by |
| have hd : ContDiff ℝ ∞ (fun z => holRealDeriv χ z) := contDiff_iff_contDiffAt.mpr |
| (fun z => contDiffAt_pi.mpr (fun a => contDiffAt_holRealDeriv hχ.contDiffAt a)) |
| have hdd : ContDiff ℝ ∞ (fun z => PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z) := |
| contDiff_iff_contDiffAt.mpr (fun z => PotentialKaehler.potentialMatrix_smooth (hχ.mul hχ).contDiffAt) |
| exact (hχ.continuous.abs.add hd.continuous.norm).add hdd.continuous.norm |
| obtain ⟨b,hb⟩ := hL.bddAbove_image hs.continuousOn |
| refine ⟨1+max 0 b,by positivity,?_⟩ |
| intro z hz |
| have hh := (hb (mem_image_of_mem _ hz)).trans (le_max_right (0:ℝ) b) |
| dsimp [size] at hh |
| refine ⟨?_,?_,?_⟩ <;> linarith [abs_nonneg (χ z),norm_nonneg (holRealDeriv χ z), |
| norm_nonneg (PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z)] |
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| theorem calabi_bound_on_compact (q : Fin A.count) {L : Set (Coordinates d)} |
| (hL : IsCompact L) (hLt : L ⊆ (A.chart q).target) |
| (χ : Coordinates d → ℝ) (hχ : ContDiff ℝ ∞ χ) |
| (hχL : tsupport χ ⊆ interior L) {M R : ℝ} (hM : 0 ≤ M) (hR : 0 ≤ R) : |
| ∃ P : ℝ, 0 < P ∧ ∀ g : KaehlerMetric A, |
| (∀ z ∈ L, ‖g.matrix q z‖ ≤ M ∧ ‖(g.matrix q z)⁻¹‖ ≤ M ∧ |
| ‖g.curvatureRicci q z‖ ≤ R ∧ ∀ a, ‖holDerivative (g.curvatureRicci q) z a‖ ≤ R) → |
| ∀ z ∈ L, χ z = 1 → g.calabiNorm q z ≤ P := by |
| obtain ⟨C,hC,hCal⟩ := calabiNorm_lower_bound d M R hM hR |
| obtain ⟨B,hB,hcut⟩ := compact_cutoff_derivative_bound hL χ hχ |
| let K := 2*(Fintype.card (TensorIndex (Fin d)):ℝ)^4*M^13 |
| let δ := (1+(Fintype.card (TensorIndex (Fin d)):ℝ)^2*M^5)⁻¹ |
| have hK : 0 ≤ K := by dsimp [K]; positivity |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| let a := cutoffCoefficient d M B C K δ |
| have ha : 0 < a := cutoffCoefficient_pos hM hB.le hC.le hK hδ |
| let Q := B^2*C+a*((d:ℝ)*R) |
| have hQ : 0 ≤ Q := by dsimp [Q]; positivity |
| refine ⟨1+B^2*Q+a*((d:ℝ)*M),by positivity,?_⟩ |
| intro g hbounds z hz hzχ |
| let f : Coordinates d → ℝ := fun y => χ y*χ y*g.calabiNorm q y+a*g.flatTrace q y |
| have hfs (y : Coordinates d) (hy : y ∈ L) : ContDiffAt ℝ ∞ f y := |
| ((hχ.contDiffAt.mul hχ.contDiffAt).mul (g.calabiNorm_smooth q (hLt hy))).add |
| (contDiffAt_const.mul (g.flatTrace_smooth q (hLt hy))) |
| obtain ⟨x,hx,hmax⟩ := hL.exists_isMaxOn ⟨z,hz⟩ (fun y hy => (hfs y hy).continuousAt.continuousWithinAt) |
| have hzle : g.calabiNorm q z ≤ f z := by |
| change _ ≤ χ z*χ z*g.calabiNorm q z+a*g.flatTrace q z |
| rw [hzχ] |
| have ht := mul_nonneg ha.le (g.flatTrace_nonneg q (hLt hz)) |
| nlinarith |
| have hmaxbound : f x ≤ B^2*Q+a*((d:ℝ)*M) := by |
| have ht := g.flatTrace_bound q x (hbounds x hx).1 |
| by_cases h0 : χ x = 0 |
| · change χ x*χ x*g.calabiNorm q x+a*g.flatTrace q x ≤ _ |
| rw [h0] |
| have hh := mul_le_mul_of_nonneg_left ht ha.le |
| nlinarith [mul_nonneg (sq_nonneg B) hQ] |
| · have hn : L ∈ nhds x := mem_interior_iff_mem_nhds.mp (hχL (subset_tsupport _ h0)) |
| have hlocal : IsLocalMax f x := Filter.eventually_of_mem hn (fun y hy => hmax hy) |
| have hnonpos := EllipticKernel.trace_potentialMatrix_nonpos_at_localMax |
| (hfs x hx) hlocal (g.matrix q x) (g.positive q x (hLt hx)) |
| have hb := hbounds x hx |
| have hgrad : ‖holRealDeriv (g.calabiNorm q) x‖^2 ≤ K*g.calabiNorm q x*g.calabiEnergy q x := |
| norm_pi_sq_bound (by positivity [g.calabiNorm_nonneg q (hLt hx),g.calabiEnergy_nonneg q (hLt hx)]) |
| (fun j => g.calabiNorm_gradient_sq_bound q (hLt hx) hM hb.1 hb.2.1 j) |
| have hlow := cutoff_test_lower_bound (g.matrix q x) (g.positive q x (hLt hx)) |
| hχ.contDiffAt (g.calabiNorm_smooth q (hLt hx)) (g.flatTrace_smooth q (hLt hx)) |
| hM hB.le hC.le hK hδ (g.calabiEnergy_nonneg q (hLt hx)) (g.calabiNorm_nonneg q (hLt hx)) |
| hb.2.1 (hcut x hx).1 (hcut x hx).2.1 (hcut x hx).2.2 hgrad |
| (hCal g q x (hLt hx) hb.1 hb.2.1 hb.2.2.1 hb.2.2.2) |
| (g.flatTrace_lower_bound q (hLt hx) hM hb.1 hb.2.1 hb.2.2.1) |
| have hS : g.calabiNorm q x ≤ Q := by |
| change g.calabiNorm q x-B^2*C-a*((d:ℝ)*R) ≤ localLap (g.matrix q x) f x at hlow |
| change localLap (g.matrix q x) f x ≤ 0 at hnonpos |
| dsimp [Q] |
| linarith |
| have hcx : χ x*χ x ≤ B^2 := by |
| have hh := (sq_le_sq₀ (abs_nonneg (χ x)) hB.le).mpr (hcut x hx).1 |
| rw [sq_abs] at hh |
| simpa only [pow_two] using hh |
| have hh := mul_le_mul hcx hS (g.calabiNorm_nonneg q (hLt hx)) (sq_nonneg B) |
| have hhT := mul_le_mul_of_nonneg_left ht ha.le |
| change χ x*χ x*g.calabiNorm q x+a*g.flatTrace q x ≤ _ |
| linarith |
| exact (hzle.trans (hmax hz)).trans (hmaxbound.trans (by linarith)) |
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| end Anticanonical.SourceSmooth.KaehlerMetric |
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| end |
| end |
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| end OAI |
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