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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import Mathlib
import OAI.Geometry.PrescribedPotential.CalabiCutoffFrames
import OAI.Geometry.PrescribedPotential.CalabiCutoffPoint
import OAI.Geometry.PrescribedPotential.CalabiFrameScalars
import OAI.Geometry.PrescribedPotential.CalabiPointInequalities
import OAI.Geometry.PrescribedPotential.CalabiRicciFrames
import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
namespace OAI
/-! Calabi Local Uniform. -/
section
noncomputable section
open Set Metric Filter Topology Matrix
open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
namespace KaehlerCalculus
variable {n : ℕ}
lemma calabi_cutoff_max_uniform {Y : Type*} [TopologicalSpace Y]
{P : Set Y} {L : Set (V n)} (G : V n → Matrix (Fin n) (Fin n) ℂ)
(H : Y × V n → Matrix (Fin n) (Fin n) ℂ) (J : Y × V n → ConnectionTensor n)
(χ : V n → ℝ) (hχ : ContDiff ℝ ∞ χ)
{R K B C δ : ℝ} (hC : 0 ≤ C) (hδ : 0 < δ)
(hframe : ∀ z ∈ L, ∀ M C D : Matrix (Fin n) (Fin n) ℂ,
((R:ℂ) • M-G z).PosSemidef → ((K:ℂ) • G z-M).PosSemidef →
C*D = 1 → Cᴴ*M*C = 1 → ‖C‖ ≤ B ∧ ‖D‖ ≤ B)
(hcut : ∀ (b : V n →L[ℂ] V n) w, b w ∈ L → ‖frameMatrix b‖ ≤ B →
-C ≤ (PotentialKaehler.potentialMatrix (fun y => (χ (b y))^2) w).trace.re ∧
holGradientSquare (fun y => (χ (b y))^2) w ≤ C*(χ (b w))^2)
(hric : ∀ x ∈ P ×ˢ L, ∀ p ∈ boundedFramePairs (n := n) B, ∀ T : ConnectionTensor n,
-C*(1+tensorSquare T) ≤ calabiResidual (p.1ᴴ*H x*p.1) (frameRicciJet (J x) p.1) T ∧
-C ≤ (p.2ᴴ*(p.1ᴴ*H x*p.1)*p.2).trace.re ∧
δ*tensorSquare T ≤ rescueSquare p.2 T)
(M : LocalKaehlerField n) {t : Y} (ht : t ∈ P) {z : V n} (hz : z ∈ L)
(hzM : z ∈ M.domain)
(hell : ((R:ℂ) • M.matrix z-G z).PosSemidef ∧ ((K:ℂ) • G z-M.matrix z).PosSemidef)
(hH : M.ricciHessian z = H (t,z))
(hJ : (fun i => mderiv (-Complex.I) (e i) M.ricciHessian z) = J (t,z))
(hχ0 : 0 ≤ χ z) (hχ1 : χ z ≤ 1)
(hmax : IsLocalMax (fun y => (χ y)^2*M.calabiNorm y+((6*C+1)/δ)*(M.matrix y).trace.re) z) :
M.calabiNorm z ≤ C*(1+(6*C+1)/δ) := by
obtain ⟨b,w,hb,hw,hnorm⟩ := normalized_frame M.matrix z (M.positive z hzM)
let D := (frameMatrix b)⁻¹
have hCD : frameMatrix b*D = 1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hb)
have hDC : D*frameMatrix b = 1 := Matrix.nonsing_inv_mul _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hb)
have hnorm' : (frameMatrix b)ᴴ*M.matrix z*frameMatrix b = 1 := by
simpa only [pullMetric,hw] using hnorm
obtain ⟨hbc,hdc⟩ := hframe z hz (M.matrix z) (frameMatrix b) D hell.1 hell.2 hCD hnorm'
have hp : (frameMatrix b,D) ∈ boundedFramePairs B := ⟨hbc,hdc,hCD,hDC⟩
have hwM : b w ∈ M.domain := hw ▸ hzM
have hwL : b w ∈ L := hw ▸ hz
have hS := M.pull_calabiNorm b hb hwM
have hS' : tensorSquare (fun i => (M.pull b hb).connection i w) = M.calabiNorm z := by
rw [← (M.pull b hb).calabiNorm_at_one hnorm,hS,hw]
obtain ⟨hr1,hr2,hr3⟩ := hric (t,z) ⟨ht,hz⟩ (frameMatrix b,D) hp (fun i => (M.pull b hb).connection i w)
have hh : (M.pull b hb).ricciHessian w = (frameMatrix b)ᴴ*H (t,z)*frameMatrix b := by
rw [M.pull_ricciHessian b hb hwM]
simp only [pullMetric,hw,hH]
have hj : (fun i => mderiv (-Complex.I) (e i) (M.pull b hb).ricciHessian w) = frameRicciJet (J (t,z)) (frameMatrix b) := by
rw [M.pull_ricciHessian_jet b hb hwM,hw,hJ]
obtain ⟨hc1,hc2⟩ := hcut b w hwL hbc
have hχs : ContDiffAt ℝ ∞ (fun y => (χ (b y))^2) w :=
(hχ.contDiffAt.comp w (b.restrictScalars ℝ).contDiff.contDiffAt).pow 2
have hχb1 : (χ (b w))^2 ≤ 1 := by rw [hw]; nlinarith
have hnormM : (M.pull b hb).matrix w = 1 := hnorm
have hhmax := M.pull_cutoff_max b hb hzM hw D hCD χ ((6*C+1)/δ) hmax
have hbound := (M.pull b hb).calabi_cutoff_max_at_one hwM hnormM D hχs hC hδ
(sq_nonneg _) hχb1 hc1 hc2
(by rw [hh,hj,(M.pull b hb).calabiNorm_at_one hnormM]; exact hr1)
(by rw [hh]; exact hr2)
(by rw [(M.pull b hb).calabiNorm_at_one hnormM]; exact hr3) hhmax
rwa [hS,hw] at hbound
theorem calabi_interior_uniform {Y : Type*} [TopologicalSpace Y]
{P : Set Y} (hP : IsCompact P) {L : Set (V n)} (hL : IsCompact L)
(G : V n → Matrix (Fin n) (Fin n) ℂ)
(hG : ContinuousOn G L) (hGp : ∀ z ∈ L, (G z).PosDef)
(H : Y × V n → Matrix (Fin n) (Fin n) ℂ) (J : Y × V n → ConnectionTensor n)
(hH : ContinuousOn H (P ×ˢ L)) (hJ : ContinuousOn J (P ×ˢ L))
(χ : V n → ℝ) (hχ : ContDiff ℝ ∞ χ)
(hχ0 : ∀ z ∈ L, 0 ≤ χ z) (hχ1 : ∀ z ∈ L, χ z ≤ 1)
(hχint : ∀ z ∈ L, χ z ≠ 0 → z ∈ interior L)
{R K : ℝ} (hR : 0 ≤ R) (hK : 0 ≤ K) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (M : LocalKaehlerField n) (t : Y),
t ∈ P → L ⊆ M.domain →
(∀ z ∈ L, ((R:ℂ) • M.matrix z-G z).PosSemidef ∧ ((K:ℂ) • G z-M.matrix z).PosSemidef) →
(∀ z ∈ L, M.ricciHessian z = H (t,z)) →
(∀ z ∈ L, (fun i => mderiv (-Complex.I) (e i) M.ricciHessian z) = J (t,z)) →
∀ z ∈ L, χ z = 1 → M.calabiNorm z ≤ C := by
obtain ⟨B,hB,hframe⟩ := compact_normalizing_frames hL G hG hGp hR hK
obtain ⟨C1,δ,hC1,hδ,hric⟩ := calabi_frame_uniform (hP.prod hL) H J hH hJ B
obtain ⟨C2,hC2,hcut⟩ := cutoff_frame_bounds hL χ hχ B
obtain ⟨Tb,hTb⟩ := hL.exists_bound_of_continuousOn (continuousOn_trace_re hG)
let C := max C1 C2
let T := K*max Tb 0
let a := (6*C+1)/δ
have hC : 0 ≤ C := hC1.trans (le_max_left _ _)
have hT : 0 ≤ T := mul_nonneg hK (le_max_right _ _)
have ha : 0 ≤ a := div_nonneg (by positivity) hδ.le
refine ⟨C*(1+a)+a*T,by positivity,?_⟩
intro M t ht hdom hell hRic hJet z hz hχz
have htr (y) (hy : y ∈ L) : (M.matrix y).trace.re ≤ T := by
exact (metric_trace_le (hell y hy).2).trans (mul_le_mul_of_nonneg_left
((le_abs_self _).trans ((hTb y hy).trans (le_max_left _ _))) hK)
let F : V n → ℝ := fun y => (χ y)^2*M.calabiNorm y+a*(M.matrix y).trace.re
have hFc : ContinuousOn F L :=
((hχ.continuous.continuousOn.pow 2).mul (M.calabiNorm_smooth.continuousOn.mono hdom)).add
(continuousOn_const.mul (continuousOn_trace_re (M.smooth.continuousOn.mono hdom)))
obtain ⟨y,hy,hmax⟩ := hL.exists_isMaxOn ⟨z,hz⟩ hFc
have hFz : M.calabiNorm z ≤ F z := by
dsimp only [F]
rw [hχz,one_pow,one_mul]
have ht0 := (Complex.nonneg_iff.mp (M.positive z (hdom hz)).posSemidef.trace_nonneg).1
exact le_add_of_nonneg_right (mul_nonneg ha ht0)
have hFzy := hmax hz
have hFbound : F y ≤ C*(1+a)+a*T := by
by_cases hzero : χ y = 0
· dsimp only [F]
rw [hzero,zero_pow (by decide),zero_mul,zero_add]
exact (mul_le_mul_of_nonneg_left (htr y hy) ha).trans (le_add_of_nonneg_left (mul_nonneg hC (by positivity)))
have hlocal : IsLocalMax F y := by
filter_upwards [mem_interior_iff_mem_nhds.mp (hχint y hy hzero)] with v hv
exact hmax hv
have hcut' (b : V n →L[ℂ] V n) w (hw : b w ∈ L) (hb : ‖frameMatrix b‖ ≤ B) := hcut b w hw hb
have hs : M.calabiNorm y ≤ C*(1+a) := by
apply calabi_cutoff_max_uniform G H J χ hχ hC hδ hframe
(fun b w hw hb => ⟨(neg_le_neg (le_max_right C1 C2)).trans (hcut' b w hw hb).1,
(hcut' b w hw hb).2.trans (mul_le_mul_of_nonneg_right (le_max_right C1 C2) (sq_nonneg _))⟩)
(fun x hx p hp U => ?_) M ht hy (hdom hy) (hell y hy) (hRic y hy) (hJet y hy)
(hχ0 y hy) (hχ1 y hy) hlocal
obtain ⟨h1,h2,h3⟩ := hric x hx p hp U
refine ⟨?_,(neg_le_neg (le_max_left C1 C2)).trans h2,h3⟩
have he : -C*(1+tensorSquare U) ≤ -C1*(1+tensorSquare U) := by
simpa only [neg_mul] using (neg_le_neg (mul_le_mul_of_nonneg_right
(le_max_left C1 C2) (add_nonneg zero_le_one (tensorSquare_nonneg U))))
exact he.trans h1
have hs0 := M.calabiNorm_nonneg (hdom hy)
have hc : (χ y)^2 ≤ 1 := by nlinarith [hχ0 y hy,hχ1 y hy]
have hh := (mul_le_mul_of_nonneg_right hc hs0).trans (by simpa using hs)
exact add_le_add hh (mul_le_mul_of_nonneg_left (htr y hy) ha)
exact hFz.trans (hFzy.trans hFbound)
end KaehlerCalculus
end
end
end OAI
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