| import Mathlib |
| import OAI.Geometry.CubicTorus.Profiles.SectionExtentBounds |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
| open Set MeasureTheory |
| open scoped Interval |
| namespace CubicTorus.Shape |
|
|
| theorem uniform_coefficient {h : ℝ} (hh : h ∈ Icc (51 / 50 : ℝ) (81 / 50 : ℝ)) : |
| min ((134 / 1000 : ℝ) * h ^ 3) (36 / 100 : ℝ) ≤ |
| ((91 / 100 : ℝ) * c * moment / Real.sqrt p₂ ^ 3) * |
| (h * (u h - (1 - meanQ) * dm)) ^ 3 := by |
| let d₀ : ℝ := (1 - meanQ) * dm |
| have hd0 : 0 ≤ d₀ := mul_nonneg (by linarith [meanQ_mem.2]) (by norm_num [dm]) |
| have hd : d₀ ≤ (6363 / 100000 : ℝ) := by |
| dsimp [d₀, dm] |
| linarith [moment_bounds.2.1] |
| have hh0 : 0 < h := by linarith [hh.1] |
| have hc := coefficient_lower |
| by_cases hbranch : 1 - 1 / Real.pi ≤ (24 / 25 : ℝ) / h |
| · have hip : 1 / Real.pi ≤ (1 / (314 / 100) : ℝ) := |
| div_le_div_of_nonneg_left (by norm_num) (by norm_num) (by linarith [Real.pi_gt_d2]) |
| have hb : (617 / 1000 : ℝ) ≤ 1 - 1 / Real.pi - d₀ := by linarith |
| have hbase : (617 / 1000 : ℝ) * h ≤ h * (u h - d₀) := by |
| dsimp [u] |
| rw [min_eq_left hbranch] |
| nlinarith [mul_le_mul_of_nonneg_left hb (le_of_lt hh0)] |
| have hp := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (617 / 1000 : ℝ) * h) hbase 3 |
| have hm := mul_le_mul_of_nonneg_left hp (by norm_num : (0 : ℝ) ≤ 574 / 1000) |
| have hmc := mul_le_mul_of_nonneg_right (le_of_lt hc) |
| (pow_nonneg (le_trans (by positivity : (0 : ℝ) ≤ (617 / 1000 : ℝ) * h) hbase) 3) |
| apply le_trans (min_le_left _ _) _ |
| change (134 / 1000 : ℝ) * h ^ 3 ≤ _ * (h * (u h - d₀)) ^ 3 |
| nlinarith only [hm, hmc, pow_pos hh0 3] |
| · have hbranch' : (24 / 25 : ℝ) / h ≤ 1 - 1 / Real.pi := le_of_not_ge hbranch |
| have hprod : h * d₀ ≤ (81 / 50 : ℝ) * (6363 / 100000 : ℝ) := |
| mul_le_mul hh.2 hd hd0 (by norm_num) |
| have hbase : (856 / 1000 : ℝ) ≤ h * (u h - d₀) := by |
| dsimp [u] |
| rw [min_eq_right hbranch'] |
| have heq : h * ((24 / 25 : ℝ) / h - d₀) = 24 / 25 - h * d₀ := by |
| field_simp |
| rw [heq] |
| linarith |
| have hp := pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 856 / 1000) hbase 3 |
| have hm := mul_le_mul_of_nonneg_left hp (by norm_num : (0 : ℝ) ≤ 574 / 1000) |
| have hmc := mul_le_mul_of_nonneg_right (le_of_lt hc) |
| (pow_nonneg (by linarith : 0 ≤ h * (u h - d₀)) 3) |
| apply le_trans (min_le_right _ _) _ |
| change (36 / 100 : ℝ) ≤ _ * (h * (u h - d₀)) ^ 3 |
| nlinarith only [hm, hmc] |
|
|
| end CubicTorus.Shape |
|
|
| namespace CubicTorus.NestingGain |
| open CubicTorus.Nesting CubicTorus.Shape |
|
|
| lemma continuousOn_weight : ContinuousOn weight (Ici 0) := by |
| exact Nesting.continuousOn_rawWeight.comp |
| (continuous_id.min continuous_const).continuousOn |
| (fun x hx => ⟨le_min hx (le_of_lt cutoff_pos), min_le_right _ _⟩) |
|
|
| lemma nesting_parameters {h δ q : ℝ} |
| (hh : h ∈ Icc (51 / 50 : ℝ) (81 / 50 : ℝ)) |
| (hδ : δ ∈ Icc 0 dm) (hq : q ∈ Icc 0 1) : |
| h * (u h - δ + δ * q) ∈ Icc 0 (24 / 25 : ℝ) ∧ |
| 0 ≤ c * δ ^ 2 * shape q ∧ 1 + c * δ ^ 2 * shape q ≤ p₂ := by |
| have hu := interval_placement hh |
| have hh0 : 0 ≤ h := by linarith [hh.1] |
| have hv0 : 0 ≤ u h - δ + δ * q := by |
| have := mul_nonneg hδ.1 hq.1 |
| linarith [hu.2.1, hδ.2] |
| have hv1 : u h - δ + δ * q ≤ u h := by |
| have := mul_le_mul_of_nonneg_left hq.2 hδ.1 |
| linarith |
| have hK0 : 0 ≤ c * δ ^ 2 * shape q := by |
| exact mul_nonneg (mul_nonneg (by norm_num [c]) (sq_nonneg δ)) (shape_nonneg hq) |
| have hK1 : c * δ ^ 2 * shape q ≤ c * dm ^ 2 * theta ^ 2 := by |
| apply mul_le_mul (mul_le_mul_of_nonneg_left |
| (pow_le_pow_left₀ hδ.1 hδ.2 2) (by norm_num [c])) (shape_max hq) |
| (shape_nonneg hq) (by norm_num [c, dm]) |
| exact ⟨⟨mul_nonneg hh0 hv0, |
| le_trans (mul_le_mul_of_nonneg_left hv1 hh0) hu.2.2.2.2.2⟩, |
| hK0, by dsimp [p₂]; linarith⟩ |
|
|
| lemma pointwise_uniform_gain {h δ q : ℝ} |
| (hh : h ∈ Icc (51 / 50 : ℝ) (81 / 50 : ℝ)) |
| (hδ : δ ∈ Icc 0 dm) (hq : q ∈ Icc 0 1) : |
| ((91 / 100 : ℝ) * c * δ ^ 2 * h ^ 3 / Real.sqrt p₂ ^ 3) * |
| ((u h - δ + δ * q) ^ 3 * shape q) ≤ |
| weight ((h * (u h - δ + δ * q)) / Real.sqrt (1 + c * δ ^ 2 * shape q)) - |
| weight (h * (u h - δ + δ * q)) := by |
| have hp := nesting_parameters hh hδ hq |
| have hlow := pointwise_weight_gain hp.1 hp.2.1 |
| apply le_trans _ hlow |
| have hnum : 0 ≤ (91 / 100 : ℝ) * (h * (u h - δ + δ * q)) ^ 3 * |
| (c * δ ^ 2 * shape q) := |
| mul_nonneg (mul_nonneg (by norm_num) (pow_nonneg hp.1.1 3)) hp.2.1 |
| have hden : (Real.sqrt (1 + c * δ ^ 2 * shape q)) ^ 3 ≤ Real.sqrt p₂ ^ 3 := |
| pow_le_pow_left₀ (Real.sqrt_nonneg _) (Real.sqrt_le_sqrt hp.2.2) 3 |
| have hden0 : 0 < (Real.sqrt (1 + c * δ ^ 2 * shape q)) ^ 3 := by |
| apply pow_pos |
| apply Real.sqrt_pos.2 |
| linarith [hp.2.1] |
| convert div_le_div_of_nonneg_left hnum hden0 hden using 1 |
| ring |
|
|
| theorem integrated_uniform_gain {h δ : ℝ} |
| (hh : h ∈ Icc (51 / 50 : ℝ) (81 / 50 : ℝ)) (hδ : δ ∈ Icc 0 dm) : |
| min ((134 / 1000 : ℝ) * h ^ 3) (36 / 100 : ℝ) * δ ^ 3 ≤ |
| δ * ∫ q in (0 : ℝ)..1, |
| (weight ((h * (u h - δ + δ * q)) / Real.sqrt (1 + c * δ ^ 2 * shape q)) - |
| weight (h * (u h - δ + δ * q))) := by |
| let v : ℝ → ℝ := fun q => u h - δ + δ * q |
| let K : ℝ → ℝ := fun q => c * δ ^ 2 * shape q |
| have hv : Continuous v := by dsimp [v]; fun_prop |
| have hK : Continuous K := by dsimp [K]; exact continuous_const.mul continuous_shape |
| have hc1 : ContinuousOn (fun q => weight ((h * v q) / Real.sqrt (1 + K q))) (Icc 0 1) := by |
| apply continuousOn_weight.comp |
| · apply (continuous_const.mul hv).continuousOn.div |
| (Real.continuous_sqrt.comp (continuous_const.add hK)).continuousOn |
| intro q hq |
| exact ne_of_gt (Real.sqrt_pos.2 (by |
| change 0 < 1 + c * δ ^ 2 * shape q |
| linarith [(nesting_parameters hh hδ hq).2.1])) |
| · intro q hq |
| exact div_nonneg (nesting_parameters hh hδ hq).1.1 (Real.sqrt_nonneg _) |
| have hc2 : ContinuousOn (fun q => weight (h * v q)) (Icc 0 1) := |
| continuousOn_weight.comp (continuous_const.mul hv).continuousOn |
| (fun q hq => (nesting_parameters hh hδ hq).1.1) |
| have hright : IntervalIntegrable (fun q => weight ((h * v q) / Real.sqrt (1 + K q)) - |
| weight (h * v q)) volume 0 1 := |
| ContinuousOn.intervalIntegrable_of_Icc (by norm_num) (hc1.sub hc2) |
| let A : ℝ := (91 / 100 : ℝ) * c * δ ^ 2 * h ^ 3 / Real.sqrt p₂ ^ 3 |
| have hleft : IntervalIntegrable (fun q => A * (v q ^ 3 * shape q)) volume 0 1 := |
| (continuous_const.mul ((hv.pow 3).mul continuous_shape)).intervalIntegrable 0 1 |
| have hmon := intervalIntegral.integral_mono_on (by norm_num : (0 : ℝ) ≤ 1) |
| hleft hright (fun q hq => pointwise_uniform_gain hh hδ hq) |
| rw [intervalIntegral.integral_const_mul] at hmon |
| have hA : 0 ≤ A := by |
| have hh0 : 0 ≤ h := by linarith [hh.1] |
| dsimp [A, c] |
| positivity |
| have hu : dm ≤ u h := by linarith [(interval_placement hh).2.1] |
| have hcubic := mul_le_mul_of_nonneg_left (cubic_shape_lower_dm hδ hu) hA |
| have hδlower := mul_le_mul_of_nonneg_left (le_trans hcubic hmon) hδ.1 |
| have hcoeff := mul_le_mul_of_nonneg_right (uniform_coefficient hh) (pow_nonneg hδ.1 3) |
| apply le_trans hcoeff |
| convert hδlower using 1 |
| dsimp [A, v, K] |
| ring |
|
|
| end CubicTorus.NestingGain |
|
|
| end |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma axialSectionCenter_complement (b : Fin 3 →Bool) (i : Fin 3) (y : TransverseProduct) : |
| (fun j ↦ y j+((1/2 : ℝ) : AxisCircle))+axialSectionCenter (fun k ↦ !b k) i= |
| y+axialSectionCenter b i := by |
| ext j |
| simp only [Pi.add_apply,axialSectionCenter] |
| by_cases hb : b (i.succAbove j)=true |
| · simp [hb] |
| · have hf : b (i.succAbove j)=false := Bool.eq_false_iff.mpr hb |
| simp only [hf,Bool.not_false,ite_true,Bool.false_eq_true,ite_false,add_zero] |
| rw [add_assoc,←AddCircle.coe_add] |
| norm_num [AddCircle.coe_period] |
|
|
| theorem isMinimizerAt_nested_edge_bound {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| {b : Fin 3 →Bool} |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| (i : Fin 3) (j : Fin 2) {L : ℝ} |
| (hfill : ∀ y : TransverseProduct,(∀ k : Fin 2,k≠j →y k=0) →2*‖y j‖<L → |
| y∈centeredAxialSlice (openRepresentative Eᶜ) (fun k ↦ !b k) i (octantHeight b i 0 : AxisCircle)) |
| {z : ℝ} (hz : z∈Icc (0 : ℝ) 1) {y : TransverseProduct} |
| (hy : y∈closure (centeredAxialSlice (openRepresentative E) b i (octantHeight b i z : AxisCircle))) |
| (hedge : ∀ k : Fin 2,k≠j →y k=((1/2 : ℝ) : AxisCircle)) : |
| 2*‖y j‖≤1-L := by |
| have hsub : centeredAxialSlice (openRepresentative E) b i (octantHeight b i z : AxisCircle)⊆ |
| centeredAxialSlice (openRepresentative E) b i (octantHeight b i 0 : AxisCircle) := |
| preimage_mono (isMinimizerAt_axialSlice_antitone hV hmin hR hd i ⟨by norm_num,by norm_num⟩ hz hz.1) |
| have hy0 := centeredAxialSlice_closure_subset (openRepresentative E) b i (octantHeight b i 0 : AxisCircle) |
| (closure_mono hsub hy) |
| by_contra hn |
| have hh : 2*‖y j+((1/2 : ℝ) : AxisCircle)‖<L := by |
| rw [axisCircle_norm_add_half] |
| linarith |
| have hyc := hfill (fun k ↦ y k+((1/2 : ℝ) : AxisCircle)) (fun k hk ↦ by |
| rw [hedge k hk,←AddCircle.coe_add] |
| norm_num [AddCircle.coe_period]) hh |
| rw [openRepresentative_compl_eq hV hmin] at hyc |
| change (torusAxialSplit i).symm ((octantHeight b i 0 : AxisCircle), |
| (fun k ↦ y k+((1/2 : ℝ) : AxisCircle))+axialSectionCenter (fun k ↦ !b k) i)∉closure (openRepresentative E) at hyc |
| rw [axialSectionCenter_complement] at hyc |
| exact hyc hy0 |
|
|
| theorem isMinimizerAt_bottom_nested_endpoint {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| (b : Fin 3 →Bool) |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {H : ℝ} |
| (hCMC : ∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → |
| (∀ m x,F (x+latticeVector m)=F x) → |
| (∫ z,tangentialDivergence F z.1 z.2 ∂(μ : Measure NormalBundle))= |
| H*(∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))) |
| (i : Fin 3) (ha : (axialAreaMarginal μ i : Measure AxisCircle)≪volume) |
| (hv : quotientMap (orientedUpperVertex b (fun j ↦ if j=i then true else false))∉openRepresentative E) |
| (hout : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun _ ↦ true))∉closure (openRepresentative E)) |
| (hin0 : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun j ↦ j==0))∈openRepresentative E) |
| (hin1 : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun j ↦ j==1))∈openRepresentative E) : |
| let s := volume.real (axialSlice (openRepresentative E) i (octantHeight b i 0 : AxisCircle)) |
| ∀ j : Fin 2,∀ z∈Icc (0 : ℝ) 1,∀ y∈closure (centeredAxialSlice (openRepresentative E) b i (octantHeight b i z : AxisCircle)), |
| (∀ k : Fin 2,k≠j →y k=((1/2 : ℝ) : AxisCircle)) → |
| 2*‖y j‖≤1-2/Real.pi*((H/2)*s-Real.sqrt ((H/2)^2*s^2-Real.pi*(1-s))) := by |
| intro s j z hz y hy hedge |
| obtain ⟨_,_,hb⟩ := isMinimizerAt_bottom_both_extents hV hE hmin hR b hd hμ hCMC i ha hv hout hin0 hin1 |
| obtain ⟨L,hL,hband,hfill,hcal⟩ := hb j |
| have he := isMinimizerAt_nested_edge_bound hV hmin hR hd i j hfill hz hy hedge |
| have hlow := Shape.extent_lower_bound hL.1 hcal |
| dsimp only [s] |
| linarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def nestingEndpointBound (h s : ℝ) : ℝ := |
| 1-2*h*Shape.t h/Real.pi+(2/Real.pi)*Real.sqrt (Real.pi+2*h^2*Shape.t h)*Real.sqrt (s-Shape.t h) |
|
|
| def nestingDelta (h s : ℝ) : ℝ := min Shape.dm (max (Shape.u h-nestingEndpointBound h s) 0) |
|
|
| lemma nestingDelta_mem (h s : ℝ) : nestingDelta h s∈Icc 0 Shape.dm := by |
| exact ⟨le_min (by norm_num [Shape.dm]) (le_max_right _ _),min_le_left _ _⟩ |
|
|
| lemma nestingDelta_gap {h s : ℝ} (hh : h∈Icc (51/50 : ℝ) (81/50 : ℝ)) |
| (hd : 0<nestingDelta h s) : max 0 (nestingEndpointBound h s)≤Shape.u h-nestingDelta h s := by |
| have hpos : 0<Shape.u h-nestingEndpointBound h s := by |
| have hm : 0 < max (Shape.u h-nestingEndpointBound h s) 0 := |
| hd.trans_le (min_le_right _ _) |
| exact (lt_max_iff.mp hm).resolve_right (lt_irrefl _) |
| have hsub : nestingDelta h s≤Shape.u h-nestingEndpointBound h s := by |
| dsimp only [nestingDelta] |
| rw [max_eq_left hpos.le] |
| exact min_le_right _ _ |
| have hplace := (Shape.interval_placement hh).2.1 |
| have hcap := (nestingDelta_mem h s).2 |
| apply max_le <;> linarith |
|
|
| lemma source_root_le_of_bottom {h s : ℝ} (hh : 0<h) (hs : 0 ≤ s) |
| (hdisc : Real.pi*(1-s)≤(h*s)^2) : Shape.t h ≤ s := by |
| have ht := Shape.t_spec hh |
| by_contra hn |
| have hst : s<Shape.t h := lt_of_not_ge hn |
| have hsq := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hs hst.le 2) (sq_nonneg h) |
| have hp := mul_lt_mul_of_pos_left hst Real.pi_pos |
| nlinarith [ht.2.2] |
|
|
| lemma nestingEndpointBound_ge_exact {h s : ℝ} (hh : 0<h) (hs : 0 ≤ s) |
| (hdisc : Real.pi*(1-s)≤(h*s)^2) : |
| 1-2/Real.pi*(h*s-Real.sqrt (h^2*s^2-Real.pi*(1-s)))≤nestingEndpointBound h s := by |
| have ht := Shape.t_spec hh |
| have hts := source_root_le_of_bottom hh hs hdisc |
| have hb := Shape.endpoint_slack_bound hh.le ht.1.le hts ht.2.2 |
| simpa only [nestingEndpointBound] using hb |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma normal_graph_axis_boundary_isolated {O : Set Torus} {a n : Cover} |
| (hn : ‖n‖=1) (i : Fin 3) (hz : ∀ k : Fin 3,k≠i →n k=0) |
| {f : Plane →ℝ} (hf : f 0=0) |
| (hchart : ∀ᶠ v in 𝓝 (0 : Cover),quotientMap (a+v)∈frontier O ↔ |
| normalGraph n f (normalPlaneCoordinates n v)=v) : |
| ∀ᶠ u in 𝓝 (a i),quotientMap (a+(u-a i) • EuclideanSpace.single i 1)∈frontier O →u=a i := by |
| have hni : n i≠0 := by |
| intro hh |
| have he : n=0 := by ext k; by_cases hk : k=i; subst k; exact hh; simpa using hz k hk |
| rw [he,norm_zero] at hn |
| norm_num at hn |
| let v : ℝ → Cover := fun u ↦ (u-a i) • EuclideanSpace.single i 1 |
| have hv (u : ℝ) : v u=((u-a i)/n i) • n := by |
| ext k |
| by_cases hk : k=i |
| · subst k |
| simp only [v,PiLp.smul_apply,PiLp.single_apply,ite_true,smul_eq_mul,mul_one] |
| field_simp |
| · simp [v,hk,hz k hk] |
| have hc : Tendsto v (𝓝 (a i)) (𝓝 (0 : Cover)) := by |
| have hcv : Continuous v := by dsimp only [v]; fun_prop |
| simpa [v] using hcv.tendsto (a i) |
| filter_upwards [hc.eventually hchart] with u hu hmem |
| have he := hu.mp hmem |
| have hzero : normalGraph n f (normalPlaneCoordinates n (v u))=0 := by |
| rw [hv,map_smul,normalPlaneCoordinates_normal hn,smul_zero] |
| change (normalPlaneLinearIsometry n) 0+f 0 • n=0 |
| rw [map_zero,hf,zero_smul,add_zero] |
| change normalGraph n f (normalPlaneCoordinates n (v u))=v u at he |
| rw [hzero] at he |
| have hh := congrArg (fun x : Cover ↦ x i) he |
| simp [v] at hh |
| linarith |
|
|
| lemma axialSectionCorner_motion (b : Fin 3 →Bool) (i : Fin 3) (t u : ℝ) (n : Fin 2 →Bool) : |
| axialSectionCorner b i u n=axialSectionCorner b i t n+(u-t) • EuclideanSpace.single i 1 := by |
| ext k |
| rcases Fin.eq_self_or_eq_succAbove i k with rfl|⟨j,rfl⟩ |
| · simp |
| · simp [Fin.succAbove_ne] |
|
|
| lemma octantHeight_endpoint_fixed (b : Fin 3 →Bool) (k : Fin 3) (c : Bool) : |
| -((octantHeight b k (if c then 1 else 0) : ℝ) : AxisCircle)= |
| ((octantHeight b k (if c then 1 else 0) : ℝ) : AxisCircle) := by |
| apply neg_eq_iff_add_eq_zero.mpr |
| rw [←AddCircle.coe_add] |
| cases hb : b k <;> cases c <;> norm_num [octantHeight,hb,AddCircle.coe_period] <;> exact ⟨(2 : ℤ),by norm_num⟩ |
|
|
| lemma axialSectionCorner_fixed (b : Fin 3 →Bool) (i : Fin 3) (t : ℝ) (n : Fin 2 →Bool) (j : Fin 2) : |
| (axisReflection (i.succAbove j)).torus (quotientMap (axialSectionCorner b i t n))= |
| quotientMap (axialSectionCorner b i t n) := by |
| ext k |
| rw [axisReflection_torus_apply] |
| split_ifs with hk |
| · subst k |
| change -((axialSectionCorner b i t n (i.succAbove j) : ℝ) : AxisCircle)=((axialSectionCorner b i t n (i.succAbove j) : ℝ) : AxisCircle) |
| rw [axialSectionCorner_other] |
| exact octantHeight_endpoint_fixed b _ _ |
| · rfl |
|
|
| theorem isMinimizerAt_corner_boundary_heights_discrete {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| (b : Fin 3 →Bool) (i : Fin 3) (n : Fin 2 →Bool) : |
| IsDiscrete {t : ℝ | quotientMap (axialSectionCorner b i t n)∈frontier (openRepresentative E)} := by |
| obtain ⟨μ,H,N,hN,hp,hμ,haN,hb,hs,hj,hu⟩ := Periodic.isMinimizerAt_H1_translation_jacobi hV hE hmin |
| apply isDiscrete_iff_forall_mem_exists_isOpen.mpr |
| intro t ht |
| let a := axialSectionCorner b i t n |
| have ha : quotientMap a∈frontier (openRepresentative E) := ht |
| have hz (k : Fin 3) (hk : k≠i) : N a k=0 := by |
| rcases Fin.eq_self_or_eq_succAbove i k with hi|⟨j,rfl⟩ |
| · exact (hk hi).elim |
| · exact isMinimizerAt_axis_fixed_normal_zero hV hE hmin _ (hR _) hμ hN hp haN a ha |
| (axialSectionCorner_fixed b i t n j) |
| obtain ⟨f,hf,hf0,hgrad,hg,hchart⟩ := isMinimizerAt_normalized_CMC_chart hV hE hmin hμ hN hp haN (fun x hx ↦ ⟨(hb x hx).1,(hb x hx).2.1⟩) a ha |
| have hlocal := normal_graph_axis_boundary_isolated (hb a ha).1.self_of_nhds i hz hf0 hchart |
| have hl : ∀ᶠ u in 𝓝 t,quotientMap (axialSectionCorner b i u n)∈frontier (openRepresentative E) →u=t := by |
| simpa only [a,axialSectionCorner_same,←axialSectionCorner_motion] using hlocal |
| obtain ⟨U,hUs,hU,htU⟩ := _root_.mem_nhds_iff.mp hl |
| refine ⟨U,hU,?_⟩ |
| ext u |
| constructor |
| · intro hu |
| exact mem_singleton_iff.mpr (hUs hu.1 hu.2) |
| · rintro rfl |
| exact ⟨htU,ht⟩ |
|
|
| theorem isMinimizerAt_corner_boundary_heights_null {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| (b : Fin 3 →Bool) (i : Fin 3) (n : Fin 2 →Bool) : |
| volume {t : ℝ | quotientMap (axialSectionCorner b i t n)∈frontier (openRepresentative E)}=0 := by |
| have hd := isMinimizerAt_corner_boundary_heights_discrete hV hE hmin hR b i n |
| exact (IsLindelof.countable_of_isDiscrete (IsLindelof.of_coe) hd).measure_zero volume |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def unitPlanarProfile (s : ℝ) : ℝ := |
| min (2*Real.sqrt (Real.pi*s)) (min 2 (2*Real.sqrt (Real.pi*(1-s)))) |
|
|
| lemma unitPlanarProfile_zero : unitPlanarProfile 0=0 := by |
| simp [unitPlanarProfile,le_min (by norm_num : (0 : ℝ)≤2) (by positivity : 0≤2*Real.sqrt Real.pi)] |
|
|
| lemma unitPlanarProfile_one : unitPlanarProfile 1=0 := by |
| simp [unitPlanarProfile,Real.sqrt_nonneg] |
|
|
| lemma unitPlanarProfile_nonneg (s : ℝ) : 0≤unitPlanarProfile s := by |
| dsimp only [unitPlanarProfile] |
| exact le_min (by positivity) (le_min (by norm_num) (by positivity)) |
|
|
| lemma open_outside_closure_of_not_frontier {X : Type*} [TopologicalSpace X] |
| {O : Set X} (ho : IsOpen O) {x : X} (hx : x∉frontier O) (hn : x∉O) : x∉closure O := by |
| intro hc |
| exact hx ⟨hc,by simpa only [ho.interior_eq] using hn⟩ |
|
|
| theorem isMinimizerAt_section_planar_profile {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| {b : Fin 3 →Bool} |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| (i : Fin 3) {t : ℝ} (ht : t∈Icc (1/2 : ℝ) 1) |
| (hc : ∀ n : Fin 2 →Bool,quotientMap (axialSectionCorner b i t n)∉frontier (openRepresentative E)) |
| (hn : volume (axialSlice (frontier (openRepresentative E)) i (t : AxisCircle))=0) : |
| ENNReal.ofReal (unitPlanarProfile (volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))))≤ |
| planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| let A := centeredAxialSlice (openRepresentative E) b i (t : AxisCircle) |
| have hr : PlanarRadialDownset A := isMinimizerAt_centered_section_radial hV hmin hR hd i _ |
| have hv : volume.real A=volume.real (axialSlice (openRepresentative E) i (t : AxisCircle)) := |
| centeredAxialSlice_volume _ b i _ |
| have hcorner (n : Fin 2 →Bool) : planarSquareCorner n∈A ↔ |
| quotientMap (axialSectionCorner b i t n)∈openRepresentative E := by |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarSquareCorner n+axialSectionCenter b i)∈openRepresentative E ↔ _ |
| rw [centered_section_corner] |
| have hout (n : Fin 2 →Bool) (hn : quotientMap (axialSectionCorner b i t n)∉openRepresentative E) : |
| quotientMap (axialSectionCorner b i t n)∉closure (openRepresentative E) := |
| open_outside_closure_of_not_frontier (isOpen_openRepresentative E) (hc n) hn |
| by_cases hnear : quotientMap (axialSectionCorner b i t (fun _ ↦ false))∈openRepresentative E |
| · by_cases hfar : quotientMap (axialSectionCorner b i t (fun _ ↦ true))∈openRepresentative E |
| · have hAu : A=univ := by |
| apply eq_univ_of_forall |
| intro y |
| apply hr _ _ (fun j ↦ ?_) ((hcorner _).mpr hfar) |
| simp only [planarSquareCorner,ite_true] |
| rw [axisCircle_norm_coe_small (by norm_num : |(1/2 : ℝ)|≤1/2)] |
| simpa using axisCircle_norm_le_half (y j) |
| have hs : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))=1 := by |
| rw [←hv,hAu]; simp |
| rw [hs,unitPlanarProfile_one,ENNReal.ofReal_zero] |
| exact bot_le |
| · by_cases h0 : quotientMap (axialSectionCorner b i t (fun j ↦ j==0))∈openRepresentative E |
| · by_cases h1 : quotientMap (axialSectionCorner b i t (fun j ↦ j==1))∈openRepresentative E |
| · obtain ⟨L,hL,hband,hcal,hp⟩ := isMinimizerAt_section_complement_calibration hV hmin hR hd i t (hout _ hfar) h0 h1 hn |
| exact (ENNReal.ofReal_le_ofReal ((min_le_right _ _).trans (min_le_right _ _))).trans hp |
| · have he0 : (fun k : Fin 2 ↦ if k=1 then false else true)=(fun k ↦ k==0) := by |
| funext k; fin_cases k <;> decide |
| have he1 : (fun k : Fin 2 ↦ if k=1 then true else false)=(fun k ↦ k==1) := by |
| funext k; fin_cases k <;> decide |
| have hp := isMinimizerAt_section_strip_perimeter hV hmin hR hd i 1 ht (by rwa [he0]) (by rw [he1]; exact hout _ h1) |
| exact (ENNReal.ofReal_le_ofReal ((min_le_right _ _).trans (min_le_left _ _))).trans (by simpa using hp) |
| · by_cases h1 : quotientMap (axialSectionCorner b i t (fun j ↦ j==1))∈openRepresentative E |
| · have he0 : (fun k : Fin 2 ↦ if k=0 then false else true)=(fun k ↦ k==1) := by |
| funext k; fin_cases k <;> decide |
| have he1 : (fun k : Fin 2 ↦ if k=0 then true else false)=(fun k ↦ k==0) := by |
| funext k; fin_cases k <;> decide |
| have hp := isMinimizerAt_section_strip_perimeter hV hmin hR hd i 0 ht (by rwa [he0]) (by rw [he1]; exact hout _ h0) |
| exact (ENNReal.ofReal_le_ofReal ((min_le_right _ _).trans (min_le_left _ _))).trans (by simpa using hp) |
| · exact (ENNReal.ofReal_le_ofReal (min_le_left _ _)).trans |
| (isMinimizerAt_section_disk_perimeter hV hmin hR hd i t hnear (hout _ h0) (hout _ h1)) |
| · have hAe : A=∅ := by |
| apply eq_empty_iff_forall_notMem.mpr |
| intro y hy |
| apply hnear |
| apply (hcorner _).mp |
| apply hr _ _ _ hy |
| intro j |
| simp only [planarSquareCorner,Bool.false_eq_true,ite_false,norm_zero] |
| exact norm_nonneg _ |
| have hs : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))=0 := by |
| rw [←hv,hAe]; simp |
| rw [hs,unitPlanarProfile_zero,ENNReal.ofReal_zero] |
| exact bot_le |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem planarTorusPerimeter_ge_weighted_derivative_flux |
| {A : Set TransverseProduct} (hA : MeasurableSet A) |
| {T : Set AxisCircle} (hT : MeasurableSet T) |
| {u v : ℝ → ℝ} (hu : ContDiff ℝ (⊤ : ℕ∞) u) (hv : ContDiff ℝ (⊤ : ℕ∞) v) |
| (pu : Function.Periodic u 1) (pv : Function.Periodic v 1) |
| (hu0 : ∀ y,0≤u y) (huv : ∀ y,(u y)^2+(v y)^2≤1) |
| {b : ℝ → ℝ} (hb : ContDiff ℝ (⊤ : ℕ∞) b) (hbi : Integrable b volume) |
| (hb0 : ∀ t,0≤b t) (hb1 : ∫ t,b t=1) |
| {r a c : ℝ} (hs : tsupport b ⊆ Icc (-r) r) (hac : a≤c) |
| (hshort : c-a+2*r<1) |
| (hin : ∀ t : ℝ,b (t-a)≠0 → ∀ y : TransverseProduct,y 0=(t : AxisCircle) → y 1∈T → y∈A) |
| (hout : ∀ t : ℝ,b (t-c)≠0 → ∀ y : TransverseProduct,y 0=(t : AxisCircle) → y∉A) |
| : ENNReal.ofReal ((∫ y in A,(real_periodic_deriv pv).lift (y 1))+ |
| 2*(∫ t in T,pu.lift t))≤planarTorusPerimeter A := by |
| let H := intervalSmoothHat b a c |
| let d := (a+c)/2 |
| have hHs : tsupport H ⊆ Ioo (d-1/2) (d+1/2) := by |
| apply (intervalSmoothHat_support hb.continuous hs hac).trans |
| intro x hx |
| dsimp only [d] |
| constructor <;> linarith [hx.1,hx.2] |
| have hHc : HasCompactSupport H := |
| IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport H) |
| (intervalSmoothHat_support hb.continuous hs hac) |
| let B : ℝ → ℝ := fun t ↦ b (t-a) |
| have hBs : tsupport B ⊆ Ioo (d-1/2) (d+1/2) := by |
| intro t ht |
| have hh := hs (tsupport_comp_subset_preimage b (continuous_id.sub continuous_const) ht) |
| change -r≤t-a ∧ t-a≤r at hh |
| dsimp only [d] |
| constructor <;> linarith [hh.1,hh.2] |
| have hBc : HasCompactSupport B := |
| IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport B) |
| (fun _ ht ↦ ⟨(hBs ht).1.le,(hBs ht).2.le⟩) |
| have hB : ContDiff ℝ (⊤ : ℕ∞) B := hb.comp (contDiff_id.sub contDiff_const) |
| let ψ : ℝ → ℝ := fun t ↦ 2*circlePeriodize H t-1 |
| have hψ : ContDiff ℝ (⊤ : ℕ∞) ψ := |
| (contDiff_const.mul (circlePeriodize_contDiff (intervalSmoothHat_contDiff hb a c) hHc)).sub contDiff_const |
| have hp : Function.Periodic ψ 1 := fun t ↦ by |
| dsimp only [ψ] |
| rw [circlePeriodize_periodic H t] |
| have hbψ (t : ℝ) : |ψ t|≤1 := by |
| have hh := circlePeriodize_unit_bounds hHs (fun t ↦ |
| ⟨intervalSmoothHat_nonneg hb.continuous hb0 hac t, |
| intervalSmoothHat_le_one hb.continuous hbi hb0 hb1 hac t⟩) t |
| apply abs_le.mpr |
| dsimp only [ψ] |
| constructor <;> linarith [hh.1,hh.2] |
| have hd (t : ℝ) (ht : t∈Icc (d-1/2) (d+1/2)) : deriv ψ t=2*(b (t-a)-b (t-c)) := by |
| have he := circlePeriodize_eventuallyEq hHc hHs ht |
| dsimp only [ψ] |
| rw [deriv_sub_const,deriv_const_mul 2 |
| ((circlePeriodize_contDiff (intervalSmoothHat_contDiff hb a c) hHc).differentiable (by simp) t), |
| he.deriv_eq,(intervalSmoothHat_hasDerivAt hb.continuous a c t).deriv] |
| let β : AxisCircle → ℝ := (circlePeriodize_periodic B).lift |
| have hβ : Continuous β := continuous_periodic_lift _ (circlePeriodize_contDiff hB hBc).continuous |
| have hβi : (∫ t,β t)=1 := by |
| rw [circlePeriodize_integral_short hBc hBs] |
| exact (integral_sub_right_eq_self b a).trans hb1 |
| have huC : Continuous (pu.lift : AxisCircle → ℝ) := continuous_periodic_lift _ hu.continuous |
| have hvC : Continuous ((real_periodic_deriv pv).lift : AxisCircle → ℝ) := |
| continuous_periodic_lift _ (hv.continuous_deriv (by simp)) |
| have hσC : Continuous ((real_periodic_deriv hp).lift : AxisCircle → ℝ) := |
| continuous_periodic_lift _ (hψ.continuous_deriv (by simp)) |
| have huL (t : AxisCircle) : 0≤pu.lift t := by |
| induction t using Quotient.inductionOn with | h t => exact hu0 t |
| have he (y : TransverseProduct) : |
| 2*β (y 0)*(T.indicator pu.lift (y 1))≤ |
| A.indicator (fun y : TransverseProduct ↦ (real_periodic_deriv hp).lift (y 0)*pu.lift (y 1)) y := by |
| let t : ℝ := ((AddCircle.equivIco (1 : ℝ) (d-1/2)) (y 0) : ℝ) |
| have ht0 := ((AddCircle.equivIco (1 : ℝ) (d-1/2)) (y 0)).property |
| have ht : t∈Icc (d-1/2) (d+1/2) := by |
| change d-1/2≤t ∧ t<d-1/2+1 at ht0 |
| constructor <;> linarith [ht0.1,ht0.2] |
| have hty : (t : AxisCircle)=y 0 := (AddCircle.equivIco (1 : ℝ) (d-1/2)).symm_apply_apply _ |
| have hBt : β (y 0)=b (t-a) := by |
| rw [←hty] |
| exact (circlePeriodize_eventuallyEq hBc hBs ht).eq_of_nhds |
| rw [hBt] |
| by_cases hy : y∈A |
| · have hc0 : b (t-c)=0 := by |
| by_contra hnc |
| exact hout t hnc y hty.symm hy |
| rw [indicator_of_mem hy] |
| have hd' : (real_periodic_deriv hp).lift (y 0)=2*b (t-a) := by |
| rw [←hty] |
| change deriv ψ t=2*b (t-a) |
| rw [hd t ht,hc0,sub_zero] |
| rw [hd'] |
| by_cases hyT : y 1∈T |
| · rw [indicator_of_mem hyT] |
| · rw [indicator_of_notMem hyT,mul_zero] |
| exact mul_nonneg (mul_nonneg (by norm_num) (hb0 _)) (huL _) |
| · rw [indicator_of_notMem hy] |
| by_cases hyT : y 1∈T |
| · have ha0 : b (t-a)=0 := by |
| by_contra hna |
| exact hy (hin t hna y hty.symm hyT) |
| rw [ha0,mul_zero,zero_mul] |
| · rw [indicator_of_notMem hyT,mul_zero] |
| have hiR : Integrable (fun y : TransverseProduct ↦ |
| (real_periodic_deriv hp).lift (y 0)*pu.lift (y 1)) := |
| ((hσC.comp (continuous_apply 0)).mul (huC.comp (continuous_apply 1))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hiL : Integrable (fun y : TransverseProduct ↦ 2*β (y 0)*T.indicator pu.lift (y 1)) := by |
| have hbase : Integrable (fun y : TransverseProduct ↦ 2*β (y 0)*pu.lift (y 1)) := |
| ((continuous_const.mul (hβ.comp (continuous_apply 0))).mul (huC.comp (continuous_apply 1))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| apply (hbase.indicator (hT.preimage (measurable_pi_apply 1))).congr |
| filter_upwards [] with y |
| by_cases hy : y 1∈T <;> simp [hy] |
| have hle := integral_mono hiL (hiR.indicator hA) he |
| have hlow : (∫ y : TransverseProduct,2*β (y 0)*T.indicator pu.lift (y 1))= |
| 2*(∫ t in T,pu.lift t) := by |
| simp_rw [mul_assoc] |
| rw [integral_const_mul,transverse_integral_mul,hβi,one_mul,integral_indicator hT] |
| rw [hlow,integral_indicator hA] at hle |
| have hvI : Integrable (fun y : TransverseProduct ↦ (real_periodic_deriv pv).lift (y 1)) := |
| (hvC.comp (continuous_apply 1)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hflux := planarTorusPerimeter_ge_product_flux A hψ hu hv hp pu pv hbψ huv |
| apply le_trans (ENNReal.ofReal_le_ofReal ?_) hflux |
| rw [integral_add hiR.integrableOn hvI.integrableOn] |
| linarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem planarTorusPerimeter_ge_open_derivative_flux |
| {A : Set TransverseProduct} (hA : IsOpen A) |
| {T : Set AxisCircle} (hT : IsCompact T) |
| {u v : ℝ → ℝ} (hu : ContDiff ℝ (⊤ : ℕ∞) u) (hv : ContDiff ℝ (⊤ : ℕ∞) v) |
| (pu : Function.Periodic u 1) (pv : Function.Periodic v 1) |
| (hu0 : ∀ y,0≤u y) (huv : ∀ y,(u y)^2+(v y)^2≤1) |
| {a c : ℝ} (hac : a≤c) (hshort : c-a<1) |
| (hin : ∀ y : TransverseProduct,y 0=(a : AxisCircle) →y 1∈T →y∈A) |
| (hout : ∀ y : TransverseProduct,y 0=(c : AxisCircle) →y∉closure A) |
| : ENNReal.ofReal ((∫ y in A,(real_periodic_deriv pv).lift (y 1))+ |
| 2*(∫ t in T,pu.lift t))≤planarTorusPerimeter A := by |
| obtain ⟨ε,hε,hεA⟩ := planar_open_partial_fiber_collar hA hT a hin |
| obtain ⟨δ,hδ,hδA⟩ := planar_open_fiber_collar isClosed_closure.isOpen_compl 0 c hout |
| let r : ℝ := min (min ε δ) (1-(c-a))/4 |
| have hr : 0<r := by dsimp [r]; positivity |
| have hrε : r<ε := by |
| have hh := (min_le_left (min ε δ) (1-(c-a))).trans (min_le_left ε δ) |
| dsimp only [r]; linarith |
| have hrδ : r<δ := by |
| have hh := (min_le_left (min ε δ) (1-(c-a))).trans (min_le_right ε δ) |
| dsimp only [r]; linarith |
| have hrc : c-a+2*r<1 := by |
| have hh := min_le_right (min ε δ) (1-(c-a)) |
| dsimp only [r]; linarith |
| let b : ContDiffBump (0 : ℝ) := ⟨r/2,r,half_pos hr,by linarith⟩ |
| apply planarTorusPerimeter_ge_weighted_derivative_flux hA.measurableSet hT.measurableSet |
| hu hv pu pv hu0 huv b.contDiff_normed b.integrable_normed (fun t ↦ b.nonneg_normed t) |
| b.integral_normed (r := r) (a := a) (c := c) _ hac hrc _ _ |
| · rw [b.tsupport_normed_eq,Real.closedBall_eq_Icc] |
| simp only [sub_eq_add_neg,zero_add] |
| exact Subset.rfl |
| · intro t ht y hy hyT |
| have hs : t-a∈Metric.ball (0 : ℝ) r := by |
| rw [←b.support_normed_eq (μ := volume)]; exact ht |
| have hab : |t-a|<r := by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hs |
| exact hεA t (hab.trans hrε) y hy hyT |
| · intro t ht y hy hya |
| have hs : t-c∈Metric.ball (0 : ℝ) r := by |
| rw [←b.support_normed_eq (μ := volume)]; exact ht |
| have hab : |t-c|<r := by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hs |
| exact hδA t (hab.trans hrδ) y hy (subset_closure hya) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma planar_closed_edge_collar {A : Set TransverseProduct} {e : ℝ} |
| (he : ∀ y∈closure A,y 1=((1/2 : ℝ) : AxisCircle) →2*‖y 0‖≤e) |
| {ε : ℝ} (hε : 0<ε) : |
| ∃ δ : ℝ,0<δ ∧ ∀ y∈closure A, |
| dist (y 1) ((1/2 : ℝ) : AxisCircle)<δ →2*‖y 0‖<e+ε := by |
| let K : Set TransverseProduct := closure A∩{y | e+ε≤2*‖y 0‖} |
| have hK : IsCompact K := (isClosed_closure.inter |
| (isClosed_le continuous_const (continuous_const.mul (continuous_apply 0).norm))).isCompact |
| have ho : IsOpen (Function.eval (1 : Fin 2) '' K)ᶜ := |
| (hK.image (continuous_apply 1)).isClosed.isOpen_compl |
| have hh : ((1/2 : ℝ) : AxisCircle)∈(Function.eval (1 : Fin 2) '' K)ᶜ := by |
| rintro ⟨y,hy,hy1⟩ |
| have hh := he y hy.1 hy1 |
| have hh' := hy.2 |
| change e+ε≤2*‖y 0‖ at hh' |
| linarith |
| obtain ⟨δ,hδ,hball⟩ := Metric.isOpen_iff.mp ho _ hh |
| refine ⟨δ,hδ,fun y hy hyd ↦ ?_⟩ |
| by_contra h |
| exact hball hyd ⟨y,⟨hy,le_of_not_gt h⟩,rfl⟩ |
|
|
| lemma planar_seam_probability_bound {A : Set TransverseProduct} |
| (hA : MeasurableSet A) {β : AxisCircle →ℝ} (hβ : Continuous β) |
| (hβ0 : ∀ t,0≤β t) (hβ1 : (∫ t,β t)=1) {r : ℝ} (hr : 0≤r) |
| (hband : ∀ y∈A,β (y 1)≠0 →2*‖y 0‖≤r) : |
| (∫ y in A,β (y 1))≤r := by |
| let B : Set AxisCircle := Metric.closedBall 0 (r/2) |
| have hB : MeasurableSet B := measurableSet_closedBall |
| have hi : Integrable (fun y : TransverseProduct ↦ β (y 1)) := |
| (hβ.comp (continuous_apply (1 : Fin 2))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hj : Integrable (fun y : TransverseProduct ↦ B.indicator (fun _ ↦ (1 : ℝ)) (y 0)*β (y 1)) := by |
| apply (hi.indicator (hB.preimage (measurable_pi_apply 0))).congr |
| filter_upwards [] with y |
| by_cases hy : y 0∈B <;> simp [hy] |
| have hle : (fun y : TransverseProduct ↦ A.indicator (fun y ↦ β (y 1)) y)≤ |
| (fun y : TransverseProduct ↦ B.indicator (fun _ ↦ (1 : ℝ)) (y 0)*β (y 1)) := by |
| intro y |
| dsimp only |
| by_cases hy : y∈A |
| · rw [indicator_of_mem hy] |
| by_cases hz : β (y 1)=0 |
| · simp [hz] |
| have hyB : y 0∈B := by |
| change dist (y 0) 0≤r/2 |
| rw [dist_zero_right] |
| linarith [hband y hy hz] |
| simp only [indicator_of_mem hyB,one_mul,le_refl] |
| · rw [indicator_of_notMem hy] |
| exact mul_nonneg (indicator_nonneg (fun _ _ ↦ zero_le_one) _) (hβ0 _) |
| have h := integral_mono (hi.indicator hA) hj hle |
| rw [integral_indicator hA,transverse_integral_mul,hβ1,mul_one,integral_indicator hB, |
| setIntegral_const,smul_eq_mul,mul_one] at h |
| apply h.trans |
| change (volume B).toReal≤r |
| rw [AddCircle.volume_closedBall] |
| have he : 2*(r/2)=r := by ring |
| rw [he,ENNReal.toReal_ofReal (le_min zero_le_one hr)] |
| exact min_le_right _ _ |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem planarTorusPerimeter_ge_strip_seam_calibration |
| {A : Set TransverseProduct} (hA : IsOpen A) |
| {T : Set AxisCircle} (hT : IsCompact T) |
| {u v : ℝ →ℝ} (hu : ContDiff ℝ (⊤ : ℕ∞) u) (hv : ContDiff ℝ (⊤ : ℕ∞) v) |
| (pu : Function.Periodic u 1) (pv : Function.Periodic v 1) |
| (hu0 : ∀ t,0≤u t) (huv : ∀ t,(u t)^2+(v t)^2≤1) |
| (hin : ∀ y : TransverseProduct,y 0=0 →y 1∈T →y∈A) |
| (hout : ∀ y : TransverseProduct,y 0=((1/2 : ℝ) : AxisCircle) →y∉closure A) |
| {β : AxisCircle →ℝ} (hβ : Continuous β) (hβ0 : ∀ t,0≤β t) |
| (hβ1 : (∫ t,β t)=1) {k r : ℝ} (hk : 0≤k) (hr : 0≤r) |
| (hd : ∀ t,deriv v t=2*k*(1-β (t : AxisCircle))) |
| (hband : ∀ y∈A,β (y 1)≠0 →2*‖y 0‖≤r) : |
| ENNReal.ofReal (2*k*(volume.real A-r)+2*(∫ t in T,pu.lift t))≤planarTorusPerimeter A := by |
| have hseam := planar_seam_probability_bound hA.measurableSet hβ hβ0 hβ1 hr hband |
| have hβi : Integrable (fun y : TransverseProduct ↦ β (y 1)) := |
| (hβ.comp (continuous_apply (1 : Fin 2))).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hd' (t : AxisCircle) : (real_periodic_deriv pv).lift t=2*k*(1-β t) := by |
| induction t using Quotient.inductionOn with | h t => exact hd t |
| have hid : (∫ y in A,(real_periodic_deriv pv).lift (y 1))= |
| 2*k*(volume.real A-∫ y in A,β (y 1)) := by |
| simp_rw [hd'] |
| rw [integral_const_mul,integral_sub (integrable_const (1 : ℝ)).integrableOn hβi.integrableOn, |
| setIntegral_const,smul_eq_mul,mul_one] |
| have hcal := planarTorusPerimeter_ge_open_derivative_flux hA hT hu hv pu pv hu0 huv |
| (a := 0) (c := 1/2) (by norm_num) (by norm_num) (by simpa only [AddCircle.coe_zero] using hin) hout |
| apply le_trans (ENNReal.ofReal_le_ofReal ?_) hcal |
| rw [hid] |
| have h := mul_le_mul_of_nonneg_left (sub_le_sub_left hseam (volume.real A)) |
| (show 0≤2*k by positivity) |
| linarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
