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59d1c80 | 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 | import Mathlib
import OAI.Geometry.IntegralFillings.Charts.Quadratic
import OAI.Geometry.IntegralFillings.Differentiation.LocalBounds
import OAI.Geometry.IntegralFillings.Charts.JacobianBound
import OAI.Geometry.IntegralFillings.Charts.CurrentContinuity
import OAI.Geometry.IntegralFillings.Charts.ScalarPieces
import OAI.Geometry.IntegralFillings.Slicing.CycleCoarea
import OAI.Geometry.IntegralFillings.Slicing.GeneralCoarea
import OAI.Geometry.IntegralFillings.Optimality.GoodRadius
import OAI.Geometry.IntegralFillings.Optimality.BallBoundary
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
namespace Optimality
lemma sphereArea_pos (n : ℕ) : 0 < sphereArea n :=
mul_pos (by positivity) (omega_pos (n+1))
lemma fillingCoefficient_pos (n : ℕ) : 0 < fillingCoefficient n := by
unfold fillingCoefficient
exact one_div_pos.mpr (mul_pos (by positivity) (Real.rpow_pos_of_pos (sphereArea_pos n) _))
lemma sharp_ball_identity {n : ℕ} (hn : 0 < n) {r : ℝ} (hr : 0 ≤ r) :
fillingCoefficient n * (sphereArea n * r^n) ^ fillingPower n = omega (n+1)*r^(n+1) := by
have hn0 : (n:ℝ) ≠ 0 := by exact_mod_cast hn.ne'
have hp : fillingPower n = 1 + 1/(n:ℝ) := by unfold fillingPower; field_simp
have hnp : (n:ℝ)*fillingPower n = ((n+1:ℕ):ℝ) := by
unfold fillingPower
push_cast
field_simp
rw [Real.mul_rpow (sphereArea_pos n).le (pow_nonneg hr _),
← Real.rpow_natCast_mul hr, hnp, Real.rpow_natCast]
rw [hp, Real.rpow_add (sphereArea_pos n),Real.rpow_one]
unfold fillingCoefficient
have hroot : (sphereArea n)^(1/(n:ℝ)) ≠ 0 :=
(Real.rpow_pos_of_pos (sphereArea_pos n) _).ne'
have hdn : (n:ℝ)+1 ≠ 0 := by positivity
calc 1 / (((n:ℝ)+1) * (sphereArea n)^(1/(n:ℝ))) *
(sphereArea n * (sphereArea n)^(1/(n:ℝ)) * r^(n+1)) =
(sphereArea n / ((n:ℝ)+1))*r^(n+1) := by field_simp
_ = omega (n+1)*r^(n+1) := by unfold sphereArea; field_simp
lemma sharp_ball_bound {n : ℕ} (hn : 0 < n) {r m : ℝ} (hr : 0 < r)
(hm : 0 ≤ m) (hmle : m ≤ sphereArea n*r^n) :
fillingCoefficient n * m^fillingPower n ≤ omega (n+1)*r^(n+1) := by
rw [← sharp_ball_identity hn hr.le]
apply mul_le_mul_of_nonneg_left _ (fillingCoefficient_pos n).le
apply Real.rpow_le_rpow hm hmle
unfold fillingPower
positivity
end Optimality
theorem coefficient_optimal (n : ℕ) (hn : 2 ≤ n) :
∃ T : IntegralCurrent (Euc (n + 1)) n,
CompactlySupported T.val ∧ IsCycle T.val ∧ 0 < mass T.val ∧
∀ S : IntegralCurrent (Euc (n + 1)) (n + 1),
CompactlySupported S.val → boundarySucc S.val = T.val →
fillingCoefficient n * (mass T.val) ^ fillingPower n ≤ mass S.val := by
obtain ⟨t,ht,hI,hM⟩ := Optimality.exists_good_inner_ball n
have hAI := Optimality.ambientBall_integral hI
let B := Optimality.ambientBall (n+1) t
let T : IntegralCurrent (Euc (n+1)) n := ⟨boundarySucc B,Optimality.integral_boundary hAI⟩
refine ⟨T,Optimality.ambientBall_boundary_compact n t hI.2.2.1,
Optimality.boundary_isCycle hAI.1, Optimality.ambientBall_boundary_mass_pos n ht hAI.2.2.1, ?_⟩
intro S hSc hSb
have hmass : mass T.val ≤ sphereArea n * (-t)^n :=
(Optimality.ambientBall_boundary_mass_le n t hI.2.2.1).trans hM
calc fillingCoefficient n * mass T.val ^ fillingPower n ≤ omega (n+1)*(-t)^(n+1) :=
Optimality.sharp_ball_bound (by omega) (neg_pos.mpr ht.2) (mass_nonneg _) hmass
_ = (volume (Metric.ball (0 : Euc (n+1)) (-t))).toReal :=
(Optimality.ball_volume_scaling (n+1) (neg_pos.mpr ht.2)).symm
_ ≤ mass S.val := Optimality.allFillings_mass_ge_ball n ht.1 S.property.1 hSb
end SharpIntegralFillings
end
end OAI
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