| import OAI.Geometry.Relativity.CKS.CKSMetricPositivity |
| import OAI.Geometry.Relativity.CKS.ConeSupport |
| import OAI.Geometry.Relativity.CKS.AreaGram |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CKSArea |
| noncomputable section |
| open Matrix MeasureTheory |
| open scoped ENNReal |
| abbrev AmbientMat := Matrix (Fin 3) (Fin 3) ℝ |
| abbrev TangentFrame := Matrix (Fin 3) (Fin 2) ℝ |
|
|
| def gram (g : AmbientMat) (T : TangentFrame) : Mat2 := Tᴴ*g*T |
|
|
| lemma gram_posDef {g : AmbientMat} (hg : g.PosDef) {T : TangentFrame} |
| (hT : Function.Injective T.mulVec) : (gram g T).PosDef := |
| hg.conjTranspose_mul_mul_same hT |
|
|
| lemma gram_difference (g p : AmbientMat) (T : TangentFrame) (c : ℝ) : |
| gram p T-c • gram g T=gram (p-c • g) T := by |
| simp only [gram,Matrix.mul_sub,Matrix.sub_mul,Matrix.mul_smul,Matrix.smul_mul] |
|
|
| lemma gram_density_lower {g p : AmbientMat} {T : TangentFrame} {c : ℝ} |
| (hg : g.PosDef) (hT : Function.Injective T.mulVec) (hc : 0 ≤ c) |
| (h : (p-c • g).PosSemidef) : c*density (gram g T) ≤ density (gram p T) := by |
| apply density_lower_bound (gram_posDef hg hT) hc |
| rw [gram_difference] |
| exact h.conjTranspose_mul_mul_same T |
|
|
| variable {α : Type*} [MeasurableSpace α] |
|
|
| def chartArea (g : α → AmbientMat) (T : α → TangentFrame) (μ : Measure α) : ℝ≥0∞ := |
| ∫⁻ x, ENNReal.ofReal (density (gram (g x) (T x))) ∂μ |
|
|
| theorem chartArea_lower {g p : α → AmbientMat} {T : α → TangentFrame} {c : ℝ} |
| (μ : Measure α) (hg : ∀ x, (g x).PosDef) (hT : ∀ x, Function.Injective (T x).mulVec) |
| (hc : 0 ≤ c) (h : ∀ x, (p x-c • g x).PosSemidef) : |
| ENNReal.ofReal c*chartArea g T μ ≤ chartArea p T μ := by |
| unfold chartArea |
| rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] |
| apply lintegral_mono; intro x |
| dsimp only |
| rw [← ENNReal.ofReal_mul hc] |
| exact ENNReal.ofReal_le_ofReal (gram_density_lower (hg x) (hT x) hc (h x)) |
|
|
| theorem full_infimum_lower {Cut : Type*} (a b : Cut → ℝ≥0∞) (c : ℝ≥0∞) |
| (h : ∀ s, c*a s ≤ b s) : c*(⨅ s, a s) ≤ ⨅ s, b s := by |
| apply le_iInf; intro s |
| calc c*(⨅ t, a t) ≤ c*a s := by gcongr; exact iInf_le a s |
| _ ≤ b s := h s |
|
|
| theorem full_chart_infimum_lower {Cut : Type*} {g p : Cut → α → AmbientMat} |
| {T : Cut → α → TangentFrame} (μ : Cut → Measure α) {c : ℝ} |
| (hg : ∀ s x, (g s x).PosDef) (hT : ∀ s x, Function.Injective (T s x).mulVec) |
| (hc : 0 ≤ c) (h : ∀ s x, (p s x-c • g s x).PosSemidef) : |
| ENNReal.ofReal c*(⨅ s, chartArea (g s) (T s) (μ s)) ≤ |
| ⨅ s, chartArea (p s) (T s) (μ s) := |
| full_infimum_lower _ _ _ (fun s => chartArea_lower (μ s) (hg s) (hT s) hc (h s)) |
|
|
| theorem real_full_chart_infimum_lower {Cut : Type*} {g p : Cut → α → AmbientMat} |
| {T : Cut → α → TangentFrame} (μ : Cut → Measure α) {c : ℝ} |
| (hg : ∀ s x, (g s x).PosDef) (hT : ∀ s x, Function.Injective (T s x).mulVec) |
| (hc : 0 ≤ c) (h : ∀ s x, (p s x-c • g s x).PosSemidef) |
| (finite : ⨅ s, chartArea (p s) (T s) (μ s) ≠ ∞) : |
| c*(⨅ s, chartArea (g s) (T s) (μ s)).toReal ≤ |
| (⨅ s, chartArea (p s) (T s) (μ s)).toReal := by |
| have hh := ENNReal.toReal_mono finite (full_chart_infimum_lower μ hg hT hc h) |
| simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal hc] using hh |
|
|
| end |
| end CKSArea |
|
|
| end |
|
|
| end OAI |
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|
