| import Mathlib | |
| import OAI.Geometry.CubicTorus.Estimates.TorusCoordinatePermute | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma local_graph_sphere_iff {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} | |
| (hg : Continuous g) (hg0 : g 0=0) {c : Cover} {R : ℝ} | |
| (hc : ⟪n,c⟫≠0) | |
| (hs : ∀ᶠ y in 𝓝 (0 : Plane),normalGraph n g y∈Metric.sphere c R) : | |
| ∀ᶠ v in 𝓝 (0 : Cover), | |
| (normalGraph n g (normalPlaneCoordinates n v)=v ↔ v∈Metric.sphere c R) := by | |
| have hct : Tendsto (normalPlaneCoordinates n) (𝓝 (0 : Cover)) (𝓝 (0 : Plane)) := by | |
| simpa only [map_zero] using (normalPlaneCoordinates n).continuous.tendsto (0 : Cover) | |
| have hnon : ∀ᶠ v in 𝓝 (0 : Cover), | |
| ⟪n,v⟫+g (normalPlaneCoordinates n v)-2*⟪n,c⟫≠0 := by | |
| have ht : ContinuousAt (fun v : Cover ↦ ⟪n,v⟫+g (normalPlaneCoordinates n v)-2*⟪n,c⟫) 0 := by fun_prop | |
| apply ht.eventually_ne | |
| simp only [inner_zero_right,map_zero,hg0,zero_add,zero_sub,neg_ne_zero,mul_ne_zero_iff] | |
| exact ⟨by norm_num,hc⟩ | |
| filter_upwards [hct.eventually hs,hnon] with v hv hnv | |
| constructor | |
| · intro he | |
| rwa [← he] | |
| · intro hvs | |
| have he : ‖v-c‖^2-‖normalGraph n g (normalPlaneCoordinates n v)-c‖^2=0 := by | |
| rw [← dist_eq_norm,← dist_eq_norm,Metric.mem_sphere.mp hvs,Metric.mem_sphere.mp hv,sub_self] | |
| rw [normal_squared_distance_difference hn,mul_eq_zero] at he | |
| exact (normalGraph_coordinates_eq_iff hn g v).mpr | |
| ((sub_eq_zero.mp (he.resolve_right hnv)).symm) | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem closed_spherical_set_contains_sphere {S : Set Cover} (hS : IsClosed S) | |
| {N : Cover → Cover} (hN : Continuous N) {t : ℝ} | |
| (hu : ∀ a∈S,‖N a‖=1) | |
| (hloc : ∀ a∈S,∀ᶠ v in 𝓝 (0 : Cover), | |
| (a+v∈S ↔ a+v∈Metric.sphere (a-t • N a) |t|) ∧ | |
| (a+v∈S → (a+v)-t • N (a+v)=a-t • N a)) | |
| {a : Cover} (ha : a∈S) (b : Cover) (hb : b∈Metric.sphere (a-t • N a) |t|) : | |
| b∈S ∧ b-t • N b=a-t • N a := by | |
| let c := a-t • N a | |
| let P : Set Cover := Metric.sphere c |t| | |
| have hPc : IsPreconnected P := isPreconnected_sphere (by rw [← Module.finrank_eq_rank]; norm_num [Cover,finrank_euclideanSpace_fin]) c |t| | |
| let : PreconnectedSpace P := isPreconnected_iff_preconnectedSpace.mp hPc | |
| let T : Set P := {x | (x : Cover)∈S ∧ (x : Cover)-t • N x=c} | |
| have hTc : IsClosed T := (hS.preimage continuous_subtype_val).inter | |
| (isClosed_eq (continuous_subtype_val.sub ((hN.comp continuous_subtype_val).const_smul t)) continuous_const) | |
| have hTo : IsOpen T := by | |
| apply isOpen_iff_mem_nhds.mpr | |
| intro x hx | |
| have ht : Tendsto (fun y : P ↦ (y : Cover)-(x : Cover)) (𝓝 x) (𝓝 (0 : Cover)) := by | |
| have hc : Continuous (fun y : P ↦ (y : Cover)-(x : Cover)) := continuous_subtype_val.sub continuous_const | |
| simpa only [sub_self] using hc.tendsto x | |
| filter_upwards [ht.eventually (hloc x hx.1)] with y hy | |
| have hadd : (x : Cover)+((y : Cover)-(x : Cover))=(y : Cover) := by abel | |
| have hys : (y : Cover)∈S := by | |
| have hyiff : (y : Cover)∈S ↔ (y : Cover)∈Metric.sphere ((x : Cover)-t • N x) |t| := by | |
| simpa only [hadd] using hy.1 | |
| apply hyiff.mpr | |
| rw [hx.2] | |
| exact y.property | |
| change (y : Cover)∈S ∧ (y : Cover)-t • N y=c | |
| refine ⟨hys,?_⟩ | |
| have he := hy.2 (by simpa only [hadd] using hys) | |
| simpa only [hadd,hx.2] using he | |
| have haP : a∈P := by | |
| change dist a (a-t • N a)=|t| | |
| rw [dist_eq_norm,sub_sub_cancel,norm_smul,hu a ha,mul_one,Real.norm_eq_abs] | |
| have hTu : T=univ := (show IsClopen T from ⟨hTc,hTo⟩).eq_univ ⟨⟨a,haP⟩,ha,rfl⟩ | |
| have hmem : (⟨b,hb⟩ : P)∈T := by rw [hTu]; trivial | |
| exact hmem | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma norm_quotientMap_eq_of_half_cube {z : Cover} (hz : ∀ i,|z i|≤1/2) : | |
| ‖quotientMap z‖=‖z‖ := by | |
| rw [PiLp.norm_eq_of_L2,PiLp.norm_eq_of_L2] | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| change ‖(z i : AddCircle (1 : ℝ))‖^2=‖z i‖^2 | |
| rw [(AddCircle.norm_coe_eq_abs_iff (1 : ℝ) (by norm_num : (1 : ℝ)≠0)).mpr | |
| (by simpa using hz i),Real.norm_eq_abs] | |
| lemma shortestLift_half_cube (x : Torus) (i : Fin 3) : |shortestLift x i|≤1/2 := by | |
| have h := UpperComparison.representative_mem_cube (x+quotientMap cubeCenter) i | |
| change |representative (x+quotientMap cubeCenter) i-(1/2)|≤1/2 | |
| exact abs_le.mpr ⟨by linarith [h.1],by linarith [h.2]⟩ | |
| theorem smallBall_exterior_preconnected {r : ℝ} (hr0 : 0≤r) (hr : r<1/2) : | |
| IsPreconnected {x : Torus | r<‖x‖} := by | |
| let S : Set Cover := Metric.sphere 0 (1/2) | |
| let T : Set Torus := quotientMap '' S | |
| have hS : IsPreconnected S := isPreconnected_sphere | |
| (by rw [← Module.finrank_eq_rank]; norm_num [Cover,finrank_euclideanSpace_fin]) 0 (1/2) | |
| have hT : IsPreconnected T := hS.image _ continuous_quotientMap.continuousOn | |
| have hTS : T⊆{x : Torus | r<‖x‖} := by | |
| rintro x ⟨z,hz,rfl⟩ | |
| have hn : ‖z‖=1/2 := by simpa only [Metric.mem_sphere,dist_zero_right] using (show z∈Metric.sphere 0 (1/2) from hz) | |
| change r<‖quotientMap z‖ | |
| rw [norm_quotientMap_eq_of_half_cube (fun i ↦ by | |
| simpa only [Real.norm_eq_abs,hn] using (PiLp.norm_apply_le z i))] | |
| simpa only [hn] using hr | |
| obtain ⟨z₀,hz₀⟩ := (NormedSpace.sphere_nonempty (x := (0 : Cover))).mpr (by norm_num : (0 : ℝ)≤1/2) | |
| apply isPreconnected_of_forall (quotientMap z₀) | |
| intro x hx | |
| change r<‖x‖ at hx | |
| have hx0 : x≠0 := by intro h; simp only [h,norm_zero] at hx; exact (not_lt_of_ge hr0) hx | |
| let z := shortestLift x | |
| have hz : 0<‖z‖ := by rw [norm_shortestLift]; exact norm_pos_iff.mpr hx0 | |
| let k : ℝ := (1/2)/‖z‖ | |
| let w : Cover := k • z | |
| have hk : 0<k := div_pos (by norm_num) hz | |
| have hw : ‖w‖=1/2 := by | |
| dsimp [w]; rw [norm_smul,Real.norm_of_nonneg hk.le]; dsimp [k]; field_simp | |
| have hwS : w∈S := by simpa only [S,Metric.mem_sphere,dist_zero_right] using hw | |
| let f : ℝ → Torus := fun t ↦ quotientMap (((1-t)+t*k) • z) | |
| have hf : Continuous f := continuous_quotientMap.comp | |
| ((continuous_const.sub continuous_id |>.add (continuous_id.mul continuous_const)).smul continuous_const) | |
| have hf0 : f 0=x := by simp [f,z,quotientMap_shortestLift] | |
| have hf1 : f 1=quotientMap w := by simp [f,w] | |
| have hfS : f '' Icc 0 1 ⊆ {x : Torus | r<‖x‖} := by | |
| rintro y ⟨t,ht,rfl⟩ | |
| have ht0 : 0≤1-t := sub_nonneg.mpr ht.2 | |
| have ha : 0≤(1-t)+t*k := add_nonneg ht0 (mul_nonneg ht.1 hk.le) | |
| have hc : ∀ i,|(((1-t)+t*k) • z) i|≤1/2 := by | |
| intro i | |
| have hzi := shortestLift_half_cube x i | |
| have hwi : |w i|≤1/2 := by simpa only [Real.norm_eq_abs,hw] using PiLp.norm_apply_le w i | |
| have he : (((1-t)+t*k) • z) i=(1-t)*z i+t*w i := by simp only [w,PiLp.smul_apply,smul_eq_mul]; ring | |
| rw [he] | |
| calc | |
| _≤|(1-t)*z i|+|t*w i| := abs_add_le _ _ | |
| _=(1-t)*|z i|+t*|w i| := by rw [abs_mul,abs_mul,abs_of_nonneg ht0,abs_of_nonneg ht.1] | |
| _≤(1-t)*(1/2)+t*(1/2) := add_le_add (mul_le_mul_of_nonneg_left hzi ht0) (mul_le_mul_of_nonneg_left hwi ht.1) | |
| _=1/2 := by ring | |
| change r<‖quotientMap (((1-t)+t*k) • z)‖ | |
| rw [norm_quotientMap_eq_of_half_cube hc,norm_smul,Real.norm_of_nonneg ha] | |
| have hkz : k*‖z‖=1/2 := by dsimp [k]; field_simp | |
| have hxz : r<‖z‖ := by simpa only [z,norm_shortestLift] using hx | |
| have hh := mul_nonneg ht0 (sub_nonneg.mpr hxz.le) | |
| have hh' := mul_nonneg ht.1 (sub_nonneg.mpr hr.le) | |
| have hh'' : 0<(1-t)*(‖z‖-r)+t*((1/2)-r) := by | |
| by_cases ht1 : t=1 | |
| · simpa only [ht1,sub_self,zero_mul,one_mul,zero_add] using sub_pos.mpr hr | |
| · have htp : 0<1-t := sub_pos.mpr (lt_of_le_of_ne ht.2 ht1) | |
| exact add_pos_of_pos_of_nonneg (mul_pos htp (sub_pos.mpr hxz)) hh' | |
| nlinarith | |
| refine ⟨T∪f '' Icc 0 1,union_subset hTS hfS,Or.inl ⟨z₀,hz₀,rfl⟩, | |
| Or.inr ⟨0,by norm_num,hf0⟩,?_⟩ | |
| exact hT.union (quotientMap w) ⟨w,hwS,rfl⟩ ⟨1,by norm_num,hf1⟩ | |
| (isPreconnected_Icc.image _ hf.continuousOn) | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem isMinimizerAt_umbilic_local_sphere {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) | |
| {H : ℝ} (hH : H≠0) | |
| (hu : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ∀ᶠ y in 𝓝 x,‖N y‖=1) | |
| (hd : ∀ x,quotientMap x∈frontier (openRepresentative E) → divergence N x=H) | |
| (hq : ∀ x,quotientMap x∈frontier (openRepresentative E) → | |
| H^2=2*coverTrace ((fderiv ℝ N x).comp (fderiv ℝ N x))) | |
| (a : Cover) (ha : quotientMap a∈frontier (openRepresentative E)) : | |
| ∀ᶠ v in 𝓝 (0 : Cover), | |
| (quotientMap (a+v)∈frontier (openRepresentative E) ↔ | |
| a+v∈Metric.sphere (a-(2/H) • N a) |2/H|) ∧ | |
| (quotientMap (a+v)∈frontier (openRepresentative E) → | |
| (a+v)-(2/H) • N (a+v)=a-(2/H) • N a) := by | |
| obtain ⟨n,f,σ,hn,hf,hf0,hσ,hm,hcov⟩ := | |
| isMinimizerAt_normal_graph_chart hV hE hmin hμ hN hp hae a ha | |
| have hx0 : normalGraph n f 0=0 := by | |
| simp only [normalGraph,hf0,zero_smul,add_zero] | |
| exact norm_eq_zero.mp (by rw [normalPlaneEmbedding_norm,norm_zero]) | |
| have hNa : N a=σ • graphSlopeNormal n (gradient f 0) := by | |
| simpa only [hx0,add_zero] using hm.self_of_nhds.1 | |
| have hJ : -σ*H≠0 := by rcases hσ with rfl|rfl <;> simpa using hH | |
| have hrati : 2/(-σ*H)= -(2/H)*σ := by | |
| rcases hσ with rfl|rfl <;> simp | |
| ring | |
| have habs : |2/(-σ*H)|=|2/H| := by | |
| rw [hrati,abs_mul,abs_neg] | |
| rcases hσ with rfl|rfl <;> norm_num | |
| have hgraph : ∀ᶠ y in 𝓝 (0 : Plane), | |
| planeDivergence (fun z ↦ cmcFlux (gradient f z)) y= -σ*H ∧ | |
| (-σ*H)^2=2*graphSquaredSecondFundamentalForm f y := by | |
| filter_upwards [hm,hm.eventually_nhds] with y hy hym | |
| have hdiv := graph_normal_extension_divergence a hn hf y hσ hN | |
| (hu _ hy.2) (hym.mono fun z hz ↦ hz.1) | |
| have hcurv := graph_normal_extension_curvature a hn hf y hσ hN | |
| (hu _ hy.2) (hym.mono fun z hz ↦ hz.1) | |
| rw [hd _ hy.2] at hdiv | |
| have heq := hq _ hy.2 | |
| rw [hcurv] at heq | |
| rcases hσ with rfl|rfl <;> constructor <;> nlinarith | |
| obtain ⟨r,hr,hgr⟩ := Metric.mem_nhds_iff.mp hgraph | |
| have hc := graph_umbilic_CMC_center hn hf hr hJ | |
| (fun y hy ↦ (hgr hy).1) (fun y hy ↦ (hgr hy).2) | |
| let c : Cover := (2/(-σ*H)) • graphSlopeNormal n (gradient f 0) | |
| have hcs : ∀ᶠ y in 𝓝 (0 : Plane),normalGraph n f y∈Metric.sphere c |2/H| := by | |
| filter_upwards [Metric.ball_mem_nhds (0 : Plane) hr] with y hy | |
| have he := hc y hy | |
| rw [hx0,zero_add] at he | |
| rw [Metric.mem_sphere,dist_eq_norm] | |
| dsimp only [c] | |
| rw [← he] | |
| simp only [sub_add_cancel_left,norm_neg,norm_smul,Real.norm_eq_abs, | |
| graphSlopeNormal_unit hn,mul_one,habs] | |
| have hcn : ⟪n,c⟫≠0 := by | |
| have hnN : ⟪n,graphSlopeNormal n (gradient f 0)⟫=(cmcWeight (gradient f 0))⁻¹ := by | |
| rw [graphSlopeNormal_weight hn,real_inner_smul_right,inner_sub_right, | |
| real_inner_self_eq_norm_sq,hn,one_pow,normalPlaneEmbedding_inner hn,sub_zero,mul_one] | |
| change ⟪n,(2/(-σ*H)) • graphSlopeNormal n (gradient f 0)⟫≠0 | |
| rw [real_inner_smul_right,hnN] | |
| exact mul_ne_zero (div_ne_zero (by norm_num) hJ) (inv_ne_zero (cmcWeight_pos _).ne') | |
| have hce : c= -(2/H) • N a := by | |
| rw [hNa,smul_smul] | |
| change (2/(-σ*H)) • _= _ | |
| rw [hrati] | |
| have hloc := local_graph_sphere_iff hn hf.continuous hf0 hcn hcs | |
| have hct : Tendsto (normalPlaneCoordinates n) (𝓝 (0 : Cover)) (𝓝 (0 : Plane)) := by | |
| simpa only [map_zero] using (normalPlaneCoordinates n).continuous.tendsto (0 : Cover) | |
| filter_upwards [hcov,hloc,hct.eventually hm, | |
| hct.eventually (Metric.ball_mem_nhds (0 : Plane) hr)] with v hv hl hm' hy | |
| constructor | |
| · rw [hv,hl,Metric.mem_sphere,Metric.mem_sphere,dist_eq_norm,dist_eq_norm,hce] | |
| rw [show a+v-(a-(2/H) • N a)=v - -(2/H) • N a by module] | |
| · intro hb | |
| have hvg := hv.mp hb | |
| have he := hc _ hy | |
| rw [hx0,zero_add,hvg,hrati] at he | |
| rw [hvg] at hm' | |
| rw [hm'.1,hNa,smul_smul,smul_smul] | |
| calc | |
| _=a+(v+(-(2/H)*σ) • graphSlopeNormal n (gradient f (normalPlaneCoordinates n v))) := by module | |
| _=a+(-(2/H)*σ) • graphSlopeNormal n (gradient f 0) := by rw [he] | |
| _=_ := by module | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem periodic_local_constant_on_preconnected {Y : Type*} {F : Cover → Y} | |
| (hp : ∀ m x,F (x+latticeVector m)=F x) | |
| {K : Set Torus} (hK : IsPreconnected K) | |
| (hloc : ∀ a : Cover,quotientMap a∈K → | |
| ∀ᶠ v in 𝓝 (0 : Cover),quotientMap (a+v)∈K → F (a+v)=F a) | |
| {a b : Cover} (ha : quotientMap a∈K) (hb : quotientMap b∈K) : F a=F b := by | |
| let G : K → Y := fun x ↦ F (representative x) | |
| have hG : IsLocallyConstant G := by | |
| apply (IsLocallyConstant.iff_eventually_eq G).mpr | |
| intro x | |
| let u := representative (x : Torus) | |
| have hu : quotientMap u∈K := by simpa only [u,quotientMap_representative] using x.property | |
| obtain ⟨r,hr,hrr⟩ := Metric.mem_nhds_iff.mp (hloc u hu) | |
| have hnear : ∀ᶠ y : K in 𝓝 x,(y : Torus)∈Metric.ball (quotientMap u) r := by | |
| apply continuous_subtype_val.continuousAt.eventually | |
| simpa only [u,quotientMap_representative] using (show ∀ᶠ y : Torus in 𝓝 (x : Torus), y∈Metric.ball (x : Torus) r from Metric.ball_mem_nhds (x : Torus) hr) | |
| filter_upwards [hnear] with y hy | |
| rw [← quotientMap_image_ball u r] at hy | |
| obtain ⟨w,hw,hwy⟩ := hy | |
| have hwu : w-u∈Metric.ball (0 : Cover) r := by | |
| simpa only [Metric.mem_ball,dist_eq_norm,sub_zero] using hw | |
| have hadd : u+(w-u)=w := by abel | |
| have he := hrr hwu (by simpa only [hadd,hwy] using y.property) | |
| rw [hadd] at he | |
| change F (representative (y : Torus))=F u | |
| rw [← hwy,periodic_representative hp] | |
| exact he | |
| let : PreconnectedSpace K := isPreconnected_iff_preconnectedSpace.mp hK | |
| have he := (⟨G,hG⟩ : LocallyConstant K Y).apply_eq_of_preconnectedSpace (⟨quotientMap a,ha⟩ : K) (⟨quotientMap b,hb⟩ : K) | |
| change F (representative (quotientMap a))=F (representative (quotientMap b)) at he | |
| simpa only [periodic_representative hp] using he | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem periodic_complete_sphere_radius_lt_half {N : Cover → Cover} | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) {t : ℝ} {c : Cover} | |
| (hs : ∀ b∈Metric.sphere c |t|,b-t • N b=c) : |t|<1/2 := by | |
| by_contra! hh | |
| let m : Fin 3 → ℤ := Pi.single 0 1 | |
| let e : Cover := latticeVector m | |
| let d : ℝ := Real.sqrt (t^2-1/4) | |
| have hd : d^2=t^2-1/4 := Real.sq_sqrt (by nlinarith [sq_abs t]) | |
| let w : Cover := EuclideanSpace.single 0 (1/2)+EuclideanSpace.single 1 d | |
| have he (i : Fin 3) : e i=if i=0 then 1 else 0 := by | |
| simp [e,m,latticeVector,Pi.single_apply] | |
| have hw : ‖w‖^2=t^2 := by | |
| rw [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_three] | |
| simp only [w,PiLp.add_apply,PiLp.single_apply] | |
| norm_num [show (2 : Fin 3)≠0 by decide,show (2 : Fin 3)≠1 by decide] | |
| nlinarith [hd] | |
| have hwe : ‖w-e‖^2=t^2 := by | |
| rw [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_three] | |
| simp only [PiLp.sub_apply,he,w,PiLp.add_apply,PiLp.single_apply] | |
| norm_num [show (2 : Fin 3)≠0 by decide,show (2 : Fin 3)≠1 by decide] | |
| nlinarith [hd] | |
| have hw' : ‖w‖=|t| := by nlinarith [norm_nonneg w,abs_nonneg t,sq_abs t] | |
| have hwe' : ‖w-e‖=|t| := by nlinarith [norm_nonneg (w-e),abs_nonneg t,sq_abs t] | |
| have hb : c+w∈Metric.sphere c |t| := by | |
| rw [Metric.mem_sphere,dist_eq_norm,add_sub_cancel_left,hw'] | |
| have hbe : c+w-e∈Metric.sphere c |t| := by | |
| rw [Metric.mem_sphere,dist_eq_norm,show c+w-e-c=w-e by abel,hwe'] | |
| have h1 := hs _ hb | |
| have h2 := hs _ hbe | |
| have hn : N (c+w-e)=N (c+w) := by | |
| have hh := hp m (c+w-e) | |
| change N (c+w-e+e)=N (c+w-e) at hh | |
| simpa only [sub_add_cancel] using hh.symm | |
| rw [hn] at h2 | |
| have hz : e=0 := by | |
| have hsub := congrArg (fun x : Cover ↦ x-(c+w-t • N (c+w))) h2 | |
| have hh : (c+w-e-t • N (c+w))-(c+w-t • N (c+w))= -e := by module | |
| rw [hh,h1,sub_self,neg_eq_zero] at hsub | |
| exact hsub | |
| have he0 := congrArg (fun v : Cover ↦ v 0) hz | |
| simp [he] at he0 | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem periodic_connected_spherical_boundary {K : Set Torus} | |
| (hK : IsClosed K) (hc : IsPreconnected K) | |
| {N : Cover → Cover} (hN : Continuous N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) {t : ℝ} | |
| (hu : ∀ a : Cover,quotientMap a∈K → ‖N a‖=1) | |
| (hloc : ∀ a : Cover,quotientMap a∈K → ∀ᶠ v in 𝓝 (0 : Cover), | |
| (quotientMap (a+v)∈K ↔ a+v∈Metric.sphere (a-t • N a) |t|) ∧ | |
| (quotientMap (a+v)∈K → (a+v)-t • N (a+v)=a-t • N a)) | |
| {a : Cover} (ha : quotientMap a∈K) : | |
| |t|<1/2 ∧ K=quotientMap '' Metric.sphere (a-t • N a) |t| := by | |
| let S : Set Cover := quotientMap ⁻¹' K | |
| have hS : IsClosed S := hK.preimage continuous_quotientMap | |
| have hfull := closed_spherical_set_contains_sphere hS hN hu hloc ha | |
| have hr : |t|<1/2 := periodic_complete_sphere_radius_lt_half hp | |
| (fun b hb ↦ (hfull b hb).2) | |
| refine ⟨hr,?_⟩ | |
| apply Subset.antisymm | |
| · intro x hx | |
| let b := representative x | |
| have hb : quotientMap b∈K := by simpa only [b,quotientMap_representative] using hx | |
| let F : Cover → Torus := fun y ↦ quotientMap (y-t • N y) | |
| have hFp : ∀ m y,F (y+latticeVector m)=F y := by | |
| intro m y | |
| simp only [F,hp] | |
| rw [show y+latticeVector m-t • N y=(y-t • N y)+latticeVector m by abel, | |
| quotientMap_add,quotientMap_latticeVector,add_zero] | |
| have he : F b=F a := periodic_local_constant_on_preconnected hFp hc | |
| (fun y hy ↦ (hloc y hy).mono fun v hv hmem ↦ congrArg quotientMap (hv.2 hmem)) hb ha | |
| obtain ⟨m,hm⟩ := same_quotient_lattice he | |
| let w := b-latticeVector m | |
| have hw : w∈Metric.sphere (a-t • N a) |t| := by | |
| have hwc : w-(a-t • N a)=t • N b := by | |
| change b-t • N b=(a-t • N a)+latticeVector m at hm | |
| dsimp only [w] | |
| calc | |
| _=(b-t • N b)-((a-t • N a)+latticeVector m)+t • N b := by module | |
| _=_ := by rw [hm,sub_self,zero_add] | |
| rw [Metric.mem_sphere,dist_eq_norm,hwc,norm_smul,hu b hb,Real.norm_eq_abs,mul_one] | |
| refine ⟨w,hw,?_⟩ | |
| simp only [w,quotientMap_sub,quotientMap_latticeVector,sub_zero,b,quotientMap_representative] | |
| · rintro x ⟨b,hb,rfl⟩ | |
| exact (hfull b hb).1 | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| theorem isMinimizerAt_mean_square_equality_spherical {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (hc : IsPreconnected (frontier (openRepresentative E))) | |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) | |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) | |
| (hp : ∀ m x,N (x+latticeVector m)=N x) | |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover) ∧ | |
| (∀ᶠ y in 𝓝 (z.1 : Cover),‖N y‖=1)) | |
| {H : ℝ} (hH : H≠0) | |
| (hd : ∀ᵐ z ∂(μ : Measure NormalBundle),divergence N z.1=H) | |
| (heq : (∫ z,coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) | |
| ∂(μ : Measure NormalBundle))=H^2*isoperimetricProfile V) : | |
| 0 < |2/H| ∧ |2/H|<1/2 ∧ ∃ c : Cover, | |
| frontier (openRepresentative E)=quotientMap '' Metric.sphere c |2/H| := by | |
| have hμU := hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hmin).symm | |
| have hbd := normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμU | |
| have hmass : (μ : Measure NormalBundle).real univ=2*isoperimetricProfile V := by | |
| rw [Measure.real,← FiniteMeasure.ennreal_mass] | |
| change (μ.mass : ℝ)=_ | |
| rw [hμ.1,isoperimetricProfile_eq_of_isMinimizerAt hmin] | |
| ring | |
| let q : NormalBundle → ℝ := fun z ↦ coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) | |
| have hqi : Integrable q (μ : Measure NormalBundle) := | |
| integrable_normal_base (normal_curvature_continuous hN) μ | |
| have hnon : ∀ᵐ z ∂(μ : Measure NormalBundle),0≤2*q z-H^2 := by | |
| filter_upwards [hbd,hae,hd] with z hzb hzu hdz | |
| have hh := isMinimizerAt_mean_curvature_bound hV hE hmin hμ hN hp | |
| (hae.mono fun _ h ↦ h.1) z.1 hzb hzu.2 | |
| rw [hdz] at hh | |
| exact sub_nonneg.mpr hh | |
| have hint : (∫ z,2*q z-H^2 ∂(μ : Measure NormalBundle))=0 := by | |
| rw [integral_sub (hqi.const_mul 2) (integrable_const _),integral_const_mul, | |
| integral_const,smul_eq_mul,hmass] | |
| change 2*(∫ z,coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) | |
| ∂(μ : Measure NormalBundle))-(2*isoperimetricProfile V)*H^2=0 | |
| rw [heq]; ring | |
| have hqae := (integral_eq_zero_iff_of_nonneg_ae hnon | |
| ((hqi.const_mul 2).sub (integrable_const _))).mp hint | |
| have hunit : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ‖N x‖=1 := by | |
| apply isMinimizerAt_periodic_ae_eq_boundary hV hE hmin hμ hN.continuous.norm | |
| continuous_const (fun m x ↦ by rw [hp]) (fun _ _ ↦ rfl) | |
| filter_upwards [hae] with z hzn | |
| rw [hzn.1,z.2.property] | |
| let R := regularizedNormal N | |
| have hR : ContDiff ℝ (⊤ : ℕ∞) R := regularizedNormal_contDiff hN | |
| have hRp : ∀ m x,R (x+latticeVector m)=R x := regularizedNormal_periodic hp | |
| have heR : ∀ᵐ z ∂(μ : Measure NormalBundle),R=ᶠ[𝓝 (z.1 : Cover)]N := by | |
| filter_upwards [hae] with z hzn | |
| exact regularizedNormal_eventuallyEq hzn.2 | |
| have hRa : ∀ᵐ z ∂(μ : Measure NormalBundle),R z.1=(z.2 : Cover) := by | |
| filter_upwards [hae,heR] with z hzn hze | |
| exact hze.eq_of_nhds.trans hzn.1 | |
| have hRu : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → | |
| ∀ᶠ y in 𝓝 x,‖R y‖=1 := by | |
| intro x hx | |
| exact regularizedNormal_unit_near hN.continuous (by rw [hunit x hx]; norm_num) | |
| have hRd : ∀ x,quotientMap x∈frontier (openRepresentative E) → divergence R x=H := by | |
| apply isMinimizerAt_periodic_ae_eq_boundary hV hE hmin hμ | |
| (continuous_divergence hR) continuous_const | |
| (fun m x ↦ divergence_periodic hRp m x) (fun _ _ ↦ rfl) | |
| filter_upwards [hd,heR] with z hdz hze | |
| rw [divergence_eq_coverTrace,hze.fderiv_eq,← divergence_eq_coverTrace] | |
| exact hdz | |
| have hRq : ∀ x,quotientMap x∈frontier (openRepresentative E) → | |
| H^2=2*coverTrace ((fderiv ℝ R x).comp (fderiv ℝ R x)) := by | |
| have hh : ∀ x,quotientMap x∈frontier (openRepresentative E) → | |
| 2*coverTrace ((fderiv ℝ R x).comp (fderiv ℝ R x))=H^2 := by | |
| apply isMinimizerAt_periodic_ae_eq_boundary hV hE hmin hμ | |
| ((normal_curvature_continuous hR).const_mul 2) continuous_const | |
| (fun m x ↦ by simp only [fderiv_periodic hRp]) (fun _ _ ↦ rfl) | |
| filter_upwards [hqae,heR] with z hqz hze | |
| rw [hze.fderiv_eq] | |
| exact sub_eq_zero.mp hqz | |
| exact fun x hx ↦ (hh x hx).symm | |
| have hloc := isMinimizerAt_umbilic_local_sphere hV hE hmin hμ hR hRp hRa hH hRu hRd hRq | |
| have hmpos : 0<(μ : Measure NormalBundle).real univ := by | |
| rw [hmass]; exact mul_pos (by norm_num) (isoperimetricProfile_pos hV) | |
| have : NeZero (μ : Measure NormalBundle) := ⟨by intro hz; simp [hz,Measure.real] at hmpos⟩ | |
| obtain ⟨z,hz⟩ := hbd.exists | |
| obtain ⟨hr,hset⟩ := periodic_connected_spherical_boundary isClosed_frontier hc | |
| hR.continuous hRp (fun x hx ↦ (hRu x hx).self_of_nhds) hloc hz | |
| exact ⟨abs_pos.mpr (div_ne_zero (by norm_num) hH),hr,z.1-(2/H) • R z.1,hset⟩ | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology | |
| namespace CubicTorus.Reduction | |
| lemma second_deriv_sub_affine {D : ℝ → ℝ} {x a b : ℝ} | |
| (hD : ContDiffAt ℝ 2 D x) : | |
| deriv (deriv (fun W ↦ D W-(a*W+b))) x=deriv (deriv D) x := by | |
| have he : deriv (fun W ↦ D W-(a*W+b)) =ᶠ[𝓝 x] (fun W ↦ deriv D W-a) := by | |
| filter_upwards [hD.eventually (by norm_num)] with W hW | |
| change deriv (D-(fun W ↦ a*W+b)) W=deriv D W-a | |
| rw [deriv_sub (hW.differentiableAt (by norm_num)) (by fun_prop)] | |
| simp | |
| rw [he.deriv_eq,deriv_sub_const] | |
| theorem UpperSupport.power_affine_jet_nonneg {I C : ℝ → ℝ} {x a b : ℝ} | |
| (hs : UpperSupport I C x) (hI : ∀ y,0≤I y) (hp : 0<I x) | |
| (he : ∀ᶠ y in 𝓝 x,I y^(3/2 : ℝ)=a*y+b) : | |
| 0≤2*C x*deriv (deriv C) x+(deriv C x)^2 := by | |
| have hCx : 0<C x := hs.contact ▸ hp | |
| let D : ℝ → ℝ := fun W ↦ C W^(3/2 : ℝ) | |
| have hD : ContDiffAt ℝ 2 D x := hs.smooth.rpow_const_of_ne hCx.ne' | |
| have hmin : IsLocalMin (fun W ↦ D W-(a*W+b)) x := by | |
| have hx := he.self_of_nhds | |
| filter_upwards [he,hs.above] with W hWe hWa | |
| change C x^(3/2 : ℝ)-(a*x+b)≤C W^(3/2 : ℝ)-(a*W+b) | |
| rw [hs.contact,hx,sub_self,← hWe] | |
| exact sub_nonneg.mpr (Real.rpow_le_rpow (hI W) hWa (by norm_num)) | |
| have hnn := second_deriv_nonneg_of_local_min hmin | |
| (hD.continuousAt.sub (by fun_prop)) | |
| rw [second_deriv_sub_affine hD] at hnn | |
| change 0≤deriv (deriv (fun W ↦ C W^(3/2 : ℝ))) x at hnn | |
| rw [second_deriv_three_halves hs.smooth hCx] at hnn | |
| have hcoef : 0<(3/4 : ℝ)*C x^(-1/2 : ℝ) := by positivity | |
| exact (mul_nonneg_iff_of_pos_left hcoef).mp hnn | |
| end CubicTorus.Reduction | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| namespace Periodic | |
| theorem isMinimizerAt_power_affine_spherical {V : ℝ} (hV : V∈Ioo 0 (1/2)) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| (hc : IsPreconnected (frontier (openRepresentative E))) {a b : ℝ} | |
| (he : ∀ᶠ W in 𝓝 V,isoperimetricProfile W^(3/2 : ℝ)=a*W+b) : | |
| ∃ (c : Cover) (r : ℝ),0<r ∧ r<1/2 ∧ | |
| frontier (openRepresentative E)=quotientMap '' Metric.sphere c r := by | |
| have hV' : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hV.2]⟩ | |
| obtain ⟨μ,H,N,C,hμ,hN,hp,hn,hCMC,hs,hd,hdd⟩ := | |
| isMinimizerAt_exact_profile_hessian hV' hE hmin | |
| have hH : 0<H := (connected_CMC_strict_sign hV' hE hmin hc hμ hCMC).1 hV.2 | |
| have hμU := hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV' hE hmin).symm | |
| have hdiv := weakCMC_normal_extension hCMC hN hp hn | |
| have hpoint : ∀ᵐ z ∂(μ : Measure NormalBundle),H^2≤ | |
| 2*coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) := by | |
| filter_upwards [normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμU, | |
| hn,hdiv] with z hz hnz hHz | |
| rw [← hHz] | |
| exact isMinimizerAt_mean_curvature_bound hV' hE hmin hμ hN hp | |
| (hn.mono fun _ h ↦ h.1) z.1 hz hnz.2 | |
| have hint := integral_mono_ae (integrable_const (H^2)) | |
| ((integrable_normal_base (normal_curvature_continuous hN) μ).const_mul 2) hpoint | |
| rw [integral_const,integral_const_mul,smul_eq_mul] at hint | |
| have hmass : (μ : Measure NormalBundle).real univ=2*isoperimetricProfile V := by | |
| rw [Measure.real,← FiniteMeasure.ennreal_mass] | |
| change (μ.mass : ℝ)=_ | |
| rw [hμ.1,isoperimetricProfile_eq_of_isMinimizerAt hmin] | |
| ring | |
| rw [hmass] at hint | |
| have hjet := hs.power_affine_jet_nonneg isoperimetricProfile_nonneg | |
| (isoperimetricProfile_pos hV') he | |
| rw [hs.contact,hd] at hjet | |
| have hge := mul_nonneg (isoperimetricProfile_nonneg V) hjet | |
| have hQ : (∫ z,coverTrace ((fderiv ℝ N z.1).comp (fderiv ℝ N z.1)) | |
| ∂(μ : Measure NormalBundle))=H^2*isoperimetricProfile V := by | |
| nlinarith [hdd] | |
| obtain ⟨hr,hr',c,hc'⟩ := isMinimizerAt_mean_square_equality_spherical hV' hE hmin hc | |
| hμ hN hp hn hH.ne' hdiv hQ | |
| exact ⟨c,|2/H|,hr,hr',hc'⟩ | |
| end Periodic | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma quotientMap_image_small_sphere (c : Cover) {r : ℝ} (hr : r≤1/2) : | |
| quotientMap '' Metric.sphere c r=Metric.sphere (quotientMap c) r := by | |
| ext x | |
| constructor | |
| · rintro ⟨z,hz,rfl⟩ | |
| have he : ‖z-c‖=r := by simpa only [Metric.mem_sphere,dist_eq_norm] using hz | |
| change dist (quotientMap z) (quotientMap c)=r | |
| rw [dist_eq_norm,← quotientMap_sub,norm_quotientMap_eq_of_half_cube] | |
| · exact he | |
| · intro i | |
| simpa only [Real.norm_eq_abs] using (PiLp.norm_apply_le (z-c) i).trans (he.le.trans hr) | |
| · intro hx | |
| refine ⟨shortestLift (x-quotientMap c)+c,?_,?_⟩ | |
| · simpa only [Metric.mem_sphere,dist_eq_norm,add_sub_cancel_right,norm_shortestLift] using hx | |
| · rw [quotientMap_add,quotientMap_shortestLift,sub_add_cancel] | |
| lemma torus_ball_preconnected (c : Torus) (r : ℝ) : IsPreconnected (Metric.ball c r) := by | |
| rw [← quotientMap_representative c,← quotientMap_image_ball] | |
| exact (convex_ball (representative c) r).isPreconnected.image _ continuous_quotientMap.continuousOn | |
| lemma torus_smallBall_exterior_preconnected (c : Torus) {r : ℝ} (hr0 : 0≤r) (hr : r<1/2) : | |
| IsPreconnected {x : Torus | r<dist x c} := by | |
| have he : (fun x : Torus ↦ x+c) '' {x : Torus | r<‖x‖}={x : Torus | r<dist x c} := by | |
| ext y | |
| simp only [mem_image,mem_ofPred_eq] | |
| constructor | |
| · rintro ⟨x,hx,rfl⟩ | |
| simpa only [dist_eq_norm,add_sub_cancel_right] using hx | |
| · intro hy | |
| exact ⟨y-c,by simpa only [dist_eq_norm] using hy,sub_add_cancel _ _⟩ | |
| rw [← he] | |
| exact (smallBall_exterior_preconnected hr0 hr).image _ (continuous_id.add continuous_const).continuousOn | |
| lemma preconnected_avoiding_frontier {X : Type*} [TopologicalSpace X] | |
| {U S : Set X} (hU : IsOpen U) (hS : IsPreconnected S) (havoid : Disjoint S (frontier U)) : | |
| S⊆U ∨ S⊆Uᶜ := by | |
| have hcov : S⊆U∪(closure U)ᶜ := by | |
| intro x hx | |
| by_cases hxu : x∈U | |
| · exact Or.inl hxu | |
| · right | |
| intro hxc | |
| exact disjoint_left.mp havoid hx (by rw [hU.frontier_eq]; exact ⟨hxc,hxu⟩) | |
| rcases hS.subset_or_subset hU isClosed_closure.isOpen_compl | |
| (disjoint_compl_right.mono_left subset_closure) hcov with h|h | |
| · exact Or.inl h | |
| · exact Or.inr (h.trans (compl_subset_compl.mpr subset_closure)) | |
| theorem isMinimizerAt_spherical_boundary_phase {V : ℝ} (hV : V∈Ioo 0 1) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| {c : Cover} {r : ℝ} (hr0 : 0<r) (hr : r<1/2) | |
| (hbd : frontier (openRepresentative E)=quotientMap '' Metric.sphere c r) : | |
| E =ᵐ[torusVolume] Metric.ball (quotientMap c) r ∨ | |
| E =ᵐ[torusVolume] (Metric.ball (quotientMap c) r)ᶜ := by | |
| let U := openRepresentative E | |
| let a := quotientMap c | |
| let B := Metric.ball a r | |
| let D := {x : Torus | r<dist x a} | |
| have hbd' : frontier U=Metric.sphere a r := hbd.trans (quotientMap_image_small_sphere c hr.le) | |
| have hBa : Disjoint B (frontier U) := by | |
| rw [hbd']; exact disjoint_left.mpr fun x hx hz ↦ (ne_of_lt hx) hz | |
| have hDa : Disjoint D (frontier U) := by | |
| rw [hbd']; exact disjoint_left.mpr fun x hx hz ↦ (ne_of_gt hx) hz | |
| have hB := preconnected_avoiding_frontier (isOpen_openRepresentative E) | |
| (torus_ball_preconnected a r) hBa | |
| have hD := preconnected_avoiding_frontier (isOpen_openRepresentative E) | |
| (torus_smallBall_exterior_preconnected a hr0.le hr) hDa | |
| have hnot : ∀ᵐ x ∂torusVolume,x∉frontier U := by | |
| rw [ae_iff] | |
| simp only [not_not] | |
| change torusVolume (frontier U)=0 | |
| exact isMinimizerAt_volume_frontier_eq_zero hV hE hmin | |
| have hpart : ∀ᵐ x ∂torusVolume,x∈B ∨ x∈D := by | |
| filter_upwards [hnot] with x hx | |
| rw [hbd'] at hx | |
| exact lt_or_gt_of_ne hx | |
| have hEU := (isMinimizerAt_openRepresentative_ae hV hE hmin).symm | |
| have hne0 : ¬E =ᵐ[torusVolume] (∅ : Set Torus) := by | |
| intro h | |
| have hz := measure_congr h | |
| rw [measure_empty,hmin.2.1] at hz | |
| exact (ENNReal.ofReal_pos.mpr hV.1).ne' hz | |
| have hne1 : ¬E =ᵐ[torusVolume] (univ : Set Torus) := by | |
| intro h | |
| have ho := measure_congr h | |
| rw [torusVolume_univ,hmin.2.1] at ho | |
| have hv := congrArg ENNReal.toReal ho | |
| rw [ENNReal.toReal_ofReal hV.1.le,ENNReal.toReal_one] at hv | |
| linarith [hV.2] | |
| rcases hB with hB|hB <;> rcases hD with hD|hD | |
| · exfalso; apply hne1 | |
| filter_upwards [hEU,hpart] with x hxu hx | |
| apply propext | |
| simp only [mem_univ,iff_true] | |
| exact hxu.mpr (hx.elim (fun hb ↦ hB hb) (fun hd ↦ hD hd)) | |
| · left | |
| filter_upwards [hEU,hpart] with x hxu hx | |
| change (x∈E)=(x∈B) | |
| apply propext | |
| exact (Iff.of_eq hxu).trans ⟨fun hu ↦ hx.resolve_right (fun hd ↦ hD hd hu),fun hb ↦ hB hb⟩ | |
| · right | |
| filter_upwards [hEU,hpart] with x hxu hx | |
| change (x∈E)=(x∈Bᶜ) | |
| apply propext | |
| exact (Iff.of_eq hxu).trans ⟨fun hu hb ↦ hB hb hu,fun hb ↦ hD (hx.resolve_left hb)⟩ | |
| · exfalso; apply hne0 | |
| filter_upwards [hEU,hpart] with x hxu hx | |
| apply propext | |
| simp only [mem_empty_iff_false,iff_false] | |
| exact fun he ↦ hx.elim (fun hb ↦ hB hb (hxu.mp he)) (fun hd ↦ hD hd (hxu.mp he)) | |
| theorem Periodic.isMinimizerAt_power_affine_ball_or_complement {V : ℝ} | |
| (hV : V∈Ioo 0 (1/2)) {E : Set Torus} (hE : MeasurableSet E) | |
| (hmin : IsMinimizerAt V E) (hc : IsPreconnected (frontier (openRepresentative E))) | |
| {a b : ℝ} (he : ∀ᶠ W in 𝓝 V,isoperimetricProfile W^(3/2 : ℝ)=a*W+b) : | |
| ∃ c : Torus,∃ r : ℝ,0<r ∧ r<1/2 ∧ | |
| (E =ᵐ[torusVolume] Metric.ball c r ∨ E =ᵐ[torusVolume] (Metric.ball c r)ᶜ) := by | |
| obtain ⟨c,r,hr,hr',hbd⟩ := isMinimizerAt_power_affine_spherical hV hE hmin hc he | |
| exact ⟨quotientMap c,r,hr,hr',isMinimizerAt_spherical_boundary_phase | |
| ⟨hV.1,by linarith [hV.2]⟩ hE hmin hr hr' hbd⟩ | |
| end CubicTorus | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal NNReal Topology RealInnerProductSpace | |
| namespace CubicTorus | |
| lemma metricBall_eq_translate (c : Torus) (r : ℝ) : | |
| Metric.ball c r=(fun x : Torus ↦ x+(-c)) ⁻¹' Metric.ball 0 r := by | |
| ext x | |
| simp only [mem_preimage,Metric.mem_ball,dist_eq_norm,sub_zero,← sub_eq_add_neg] | |
| lemma metricBall_volume (c : Torus) {r : ℝ} (hr0 : 0≤r) (hr : r≤1/2) : | |
| torusVolume (Metric.ball c r)=ENNReal.ofReal (r^3*(Real.pi*4/3)) := by | |
| rw [metricBall_eq_translate,volume_translation Metric.isOpen_ball.measurableSet.nullMeasurableSet, | |
| UpperComparison.volume_centeredBall hr0 hr] | |
| lemma metricBall_perimeter (c : Torus) {r : ℝ} (hr0 : 0<r) (hr : r<1/2) : | |
| perimeter (Metric.ball c r)=ENNReal.ofReal (4*Real.pi*r^2) := by | |
| rw [metricBall_eq_translate,perimeter_translation Metric.isOpen_ball.measurableSet.nullMeasurableSet, | |
| UpperComparison.centeredBall_eq_cubeBall_preimage, | |
| perimeter_translation (UpperComparison.measurableSet_cubeBall r).nullMeasurableSet] | |
| exact perimeter_cubeBall_eq hr0 hr | |
| lemma small_minimizer_ball_phase {V : ℝ} (hV : V∈Ioc 0 (1/2)) | |
| {E : Set Torus} (hmin : IsMinimizerAt V E) {c : Torus} {r : ℝ} | |
| (hr0 : 0<r) (hr : r<1/2) | |
| (hph : E =ᵐ[torusVolume] Metric.ball c r ∨ E =ᵐ[torusVolume] (Metric.ball c r)ᶜ) : | |
| E =ᵐ[torusVolume] Metric.ball c r := by | |
| rcases hph with h|h | |
| · exact h | |
| exfalso | |
| obtain ⟨F,hF,hFv,hFp⟩ := exists_bounded_competitor (V := V) ⟨hV.1.le,by linarith [hV.2]⟩ | |
| have hP := (hmin.2.2 F hF hFv).trans hFp | |
| have hpbe : perimeter E=ENNReal.ofReal (4*Real.pi*r^2) := | |
| (perimeter_congr_ae h).trans ((perimeter_compl | |
| (Metric.isOpen_ball.measurableSet.nullMeasurableSet : NullMeasurableSet (Metric.ball c r) torusVolume)).trans | |
| (metricBall_perimeter c hr0 hr)) | |
| rw [hpbe] at hP | |
| have hP' : 4*Real.pi*r^2≤2 := by | |
| apply (ENNReal.ofReal_le_ofReal_iff (by norm_num : (0 : ℝ)≤2)).mp | |
| simpa only [ENNReal.ofReal_ofNat] using hP | |
| have hv := congrArg ENNReal.toReal (measure_congr h) | |
| have hvb : torusVolume.real (Metric.ball c r)=r^3*(Real.pi*4/3) := by | |
| rw [Measure.real,metricBall_volume c hr0.le hr.le,ENNReal.toReal_ofReal (by positivity)] | |
| change (torusVolume E).toReal=torusVolume.real (Metric.ball c r)ᶜ at hv | |
| rw [hmin.2.1,ENNReal.toReal_ofReal hV.1.le,measureReal_compl Metric.isOpen_ball.measurableSet, | |
| hvb,Measure.real,torusVolume_univ,ENNReal.toReal_one] at hv | |
| have hh := mul_le_mul_of_nonneg_right hP' hr0.le | |
| nlinarith only [hv,hh,hV.2,hr] | |
| theorem isMinimizerAt_spherical_standard_ball {V : ℝ} (hV : V∈Ioc 0 (1/2)) | |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) | |
| {c : Cover} {r : ℝ} (hr0 : 0<r) (hr : r<1/2) | |
| (hbd : frontier (openRepresentative E)=quotientMap '' Metric.sphere c r) : | |
| V≤4*Real.pi/81 ∧ CongruentAE E (canonicalBall V) := by | |
| have hV' : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hV.2]⟩ | |
| have hae := small_minimizer_ball_phase hV hmin hr0 hr | |
| (isMinimizerAt_spherical_boundary_phase hV' hE hmin hr0 hr hbd) | |
| have hv := congrArg ENNReal.toReal (measure_congr hae) | |
| rw [hmin.2.1,metricBall_volume (quotientMap c) hr0.le hr.le, | |
| ENNReal.toReal_ofReal hV.1.le,ENNReal.toReal_ofReal (by positivity)] at hv | |
| obtain ⟨hs0,hs,hsv⟩ := UpperComparison.canonicalBall_radius_properties hV | |
| have hrs : r=(3*V/(4*Real.pi))^(1/3 : ℝ) := by | |
| have he : r^3=((3*V/(4*Real.pi))^(1/3 : ℝ))^3 := by | |
| nlinarith [Real.pi_pos] | |
| exact (show Odd (3 : ℕ) by decide).strictMono_pow.injective he | |
| have hP := isMinimizerAt_perimeter_le_candidateProfile hV' hmin | |
| have hP' : 4*Real.pi*r^2≤2*Real.sqrt (Real.pi*V) := by | |
| rw [perimeter_congr_ae hae,metricBall_perimeter (quotientMap c) hr0 hr] at hP | |
| have hc : candidateProfile V≤2*Real.sqrt (Real.pi*V) := by | |
| unfold candidateProfile | |
| rw [min_eq_left (by linarith [hV.2] : V≤1-V)] | |
| exact (min_le_right _ _).trans (min_le_left _ _) | |
| exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp | |
| (hP.trans (ENNReal.ofReal_le_ofReal hc)) | |
| have hrthird : r≤1/3 := by | |
| have hh := sq_le_sq₀ (by positivity : 0≤4*Real.pi*r^2) (by positivity : 0≤2*Real.sqrt (Real.pi*V)) |>.mpr hP' | |
| have hsq := Real.sq_sqrt (show 0≤Real.pi*V from mul_nonneg Real.pi_pos.le hV.1.le) | |
| have hfactor : 0<16*Real.pi^2*r^3 := by positivity | |
| have hsq' : (4*Real.pi*r^2)^2≤4*Real.pi*V := by nlinarith only [hh,hsq] | |
| rw [hv] at hsq' | |
| have hprod : (16*Real.pi^2*r^3)*(3*r-1)≤0 := by nlinarith only [hsq'] | |
| have hh' : 3*r-1≤0 := by | |
| by_contra h | |
| exact (not_lt_of_ge hprod) (mul_pos hfactor (lt_of_not_ge h)) | |
| linarith | |
| refine ⟨?_,?_⟩ | |
| · have hc := pow_le_pow_left₀ hr0.le hrthird 3 | |
| have hp := mul_le_mul_of_nonneg_right hc (show 0≤Real.pi*4/3 by positivity) | |
| nlinarith [hv] | |
| · refine ⟨IsometryEquiv.addRight (quotientMap c),hae.trans ?_⟩ | |
| apply Filter.Eventually.of_forall | |
| intro x | |
| apply propext | |
| change x∈Metric.ball (quotientMap c) r ↔ x∈(IsometryEquiv.addRight (quotientMap c)) '' Metric.ball 0 ((3*V/(4*Real.pi))^(1/3 : ℝ)) | |
| rw [← hrs] | |
| constructor | |
| · intro hx | |
| refine ⟨x-quotientMap c,?_,by simp⟩ | |
| simpa only [Metric.mem_ball,dist_eq_norm,sub_zero] using hx | |
| · rintro ⟨y,hy,rfl⟩ | |
| simpa only [IsometryEquiv.addRight_apply,Metric.mem_ball,dist_eq_norm,add_sub_cancel_right,sub_zero] using hy | |
| end CubicTorus | |
| end | |
| end | |
| end | |
| end OAI | |