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9655094 | 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 | import OAI.Analysis.CircleDomains.Modulus.LinePayment
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.Order.Compact
namespace OAI
/-!
# Collapse forced by barriers of vanishing energy
The line-payment inequality prevents barriers of vanishing energy from
crossing every horizontal line over a fixed interval of positive length.
Applied to both coordinate projections, this forces a compact connected
planar set to be a singleton. The hypotheses specify barrier existence,
crossings, and absolute continuity of the restrictions.
-/
open MeasureTheory Set Filter
open scoped Topology
namespace CircleDomainRigidity
/-- A sequence of actual rectangle barriers, each crossing almost every
horizontal line over `(a,b]`, whose squared horizontal derivative energies
tend to zero. The supporting horizontal interval is fixed for the sequence. -/
def HasVanishingRectangleBarriers (a b : ℝ) : Prop :=
∃ L : ℝ, 0 < L ∧ ∃ v : ℕ → ℝ → ℝ → ℝ,
(∀ n, Integrable (fun p : ℝ × ℝ => deriv (v n p.1) p.2 ^ 2)
((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc (-L) L)))) ∧
(∀ n, ∀ᵐ y ∂volume.restrict (Ioc a b),
AbsolutelyContinuousOnInterval (v n y) (-L) L ∧ v n y (-L) = 0 ∧
∃ c ∈ Icc (-L) L, v n y c = 1) ∧
Tendsto (fun n => ∫ p : ℝ × ℝ, deriv (v n p.1) p.2 ^ 2
∂((volume.restrict (Ioc a b)).prod (volume.restrict (Ioc (-L) L))))
atTop (𝓝 0)
/-- A nondegenerate interval cannot support crossings by barriers whose
energies tend to zero. This is the contradiction step after line payment. -/
theorem eq_of_vanishing_rectangle_barriers {a b : ℝ} (hab : a ≤ b)
(hbar : HasVanishingRectangleBarriers a b) : a = b := by
obtain ⟨L, hL, v, hEnergy, hSlices, hlimit⟩ := hbar
have hbound : (b - a) / (2 * L) ≤ 0 :=
ge_of_tendsto hlimit (Filter.Eventually.of_forall (fun n =>
integrated_line_payment_of_absolutelyContinuous hL hab (hEnergy n) (hSlices n)))
have hlength : b - a ≤ 0 := by
simpa using (div_le_iff₀ (mul_pos (by norm_num) hL)).mp hbound
linarith
/-- If both coordinate projection intervals of a compact connected planar
set admit the barriers above, then every two points of the set coincide.
Compactness and connectedness identify the projections with the full
closed intervals between their extrema. -/
theorem subsingleton_of_vanishing_projection_barriers {K : Set ℂ}
(hcompact : IsCompact K) (hconnected : IsConnected K)
(hre : HasVanishingRectangleBarriers
(sInf (Complex.re '' K)) (sSup (Complex.re '' K)))
(him : HasVanishingRectangleBarriers
(sInf (Complex.im '' K)) (sSup (Complex.im '' K))) :
K.Subsingleton := by
have hReInterval : Complex.re '' K =
Icc (sInf (Complex.re '' K)) (sSup (Complex.re '' K)) :=
eq_Icc_of_connected_compact
(hconnected.image Complex.re Complex.continuous_re.continuousOn)
(hcompact.image Complex.continuous_re)
have hImInterval : Complex.im '' K =
Icc (sInf (Complex.im '' K)) (sSup (Complex.im '' K)) :=
eq_Icc_of_connected_compact
(hconnected.image Complex.im Complex.continuous_im.continuousOn)
(hcompact.image Complex.continuous_im)
have hReNonempty :
(Icc (sInf (Complex.re '' K)) (sSup (Complex.re '' K))).Nonempty :=
hReInterval ▸ hconnected.nonempty.image Complex.re
have hImNonempty :
(Icc (sInf (Complex.im '' K)) (sSup (Complex.im '' K))).Nonempty :=
hImInterval ▸ hconnected.nonempty.image Complex.im
have hReEq := eq_of_vanishing_rectangle_barriers (nonempty_Icc.mp hReNonempty) hre
have hImEq := eq_of_vanishing_rectangle_barriers (nonempty_Icc.mp hImNonempty) him
intro z hz w hw
have hzre : z.re ∈ Icc (sInf (Complex.re '' K)) (sSup (Complex.re '' K)) :=
hReInterval ▸ mem_image_of_mem Complex.re hz
have hwre : w.re ∈ Icc (sInf (Complex.re '' K)) (sSup (Complex.re '' K)) :=
hReInterval ▸ mem_image_of_mem Complex.re hw
have hzim : z.im ∈ Icc (sInf (Complex.im '' K)) (sSup (Complex.im '' K)) :=
hImInterval ▸ mem_image_of_mem Complex.im hz
have hwim : w.im ∈ Icc (sInf (Complex.im '' K)) (sSup (Complex.im '' K)) :=
hImInterval ▸ mem_image_of_mem Complex.im hw
apply Complex.ext
· exact le_antisymm (hzre.2.trans (hReEq ▸ hwre.1))
(hwre.2.trans (hReEq ▸ hzre.1))
· exact le_antisymm (hzim.2.trans (hImEq ▸ hwim.1))
(hwim.2.trans (hImEq ▸ hzim.1))
end CircleDomainRigidity
end OAI
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