Symplectic area on the truncated thrice-punctured sphere

Let (V, ω) be a symplectic vector space and let u be a C² map from the thrice-punctured sphere S = ℂ ∖ {0, 1} to V. The truncation S_R (TauCeti.ThricePuncturedSphere.truncation) is bounded by the three circles ±s = R of the cylindrical ends, and in the end coordinates z = s + i t of the end q the restriction of u to the circle ±s = R is the loop ℓ_q(t) = u(φ_q(±R + i t)), where φ_q is the end lift Puncture.cylindricalEndLift. This file proves Stokes' formula for the symplectic area of u over S_R:

∫_{S_R} u^*ω = a(ℓ_0) + a(ℓ_1) - a(ℓ_∞)

for R ≥ 1, where a is the loop area TauCeti.SymplecticForm.loopArea, the integral of the primitive λₓ(v) = ½ ω(x, v) of ω (SymplecticForm.integral_fderiv_apply_truncation). All three loops run clockwise in the plane. This is the boundary orientation of S_R on the two inner circles, about 0 and 1, and the reverse of it on the outer circle, which gives the signs of the ends: + on the input ends 0 and 1, - on the output end ∞.

The domain S_R is a disc with two holes, so the annulus formula TauCeti.SymplecticForm.integral_fderiv_apply_annulus does not apply to it directly. Instead the map is cut off near the puncture at 1: the product uc = (1 - χ) u with a bump function χ about 1 is C² on ℂ ∖ {0}, so the annulus formula applies to uc on the annulus about 0 bounded by the circles at 0 and ∞. The annulus formula for uc and for u on two annuli about 1 then removes the contribution of the disc about 1. No extension of u itself across the punctures is needed.

In the energy identity of the pair-of-pants product, adding the Hamiltonian terms turns these loop areas into the Hamiltonian actions of the end loops; see TauCeti.Geometry.Symplectic.Floer.PairOfPants.Action.

References

• D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., AMS Colloquium Publications 52, 2012, Section 4.5.

theorem TauCeti.SymplecticForm.integral_fderiv_apply_truncation {V : Type u_1} [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] (ω : SymplecticForm V) {u : ℂ → V} (hu : ContDiffOn ℝ 2 u {w : ℂ | w ≠ 0 ∧ w ≠ 1}) {R : ℝ} (hR : 1 ≤ R) : ∫ (w : ℂ) in ThricePuncturedSphere.truncation R, (fun (v w : V) => (ω.toBilinForm v) w) ((fderiv ℝ u w) 1) ((fderiv ℝ u w) Complex.I) = ∑ q : ThricePuncturedSphere.Puncture, q.endSign * ω.loopArea fun (t : ℝ) => u (q.cylindricalEndLift (↑(q.endSign * R) + ↑t * Complex.I))

Stokes' formula for symplectic area on the truncated thrice-punctured sphere. Let u be C² on S = ℂ ∖ {0, 1}. For R ≥ 1 the symplectic area of u over the truncation S_R is the signed sum of the loop areas of the restrictions of u to the three boundary circles ±s = R, parametrized by the end coordinates:

∫_{S_R} u^*ω = ∑ q, ±a(t ↦ u(φ_q(±R + i t))),

with the sign + on the input ends 0 and 1 and - on the output end ∞.