| 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 |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
