Curvature and energy of the pair-of-pants perturbation on its ends #
The energy of a solution of the perturbed Cauchy–Riemann equation (du - X_K)^{0,1} = 0 is its
symplectic area, plus the boundary integral of K(u), minus the integral of the curvature
R_K = ∂_s K_t - ∂_t K_s - {K_s, K_t} (TauCeti.SymplecticForm.integral_floerEnergyDensity).
An energy bound for the pair-of-pants product therefore needs a bound on ∫ sup_x |R_K| which
does not depend on how far out the ends are cut off.
This file computes the curvature of the perturbation TauCeti.Floer.pairOfPantsPerturbation in
the end coordinates z = (s, t) of each end q. The curvature formula holds for Hamiltonians on
any manifold with a two-form, and the integrated bounds are stated for a symplectic vector space.
There the perturbation is the continuation datum β(±s - 1) H_q(t) dt, where β is the smooth
transition function, so its curvature is ±β'(±s - 1) H_q(t). It is supported in the collar
1 ≤ ±s ≤ 2, and since β' ≥ 0 has total integral 1, the integrated curvature over any part of
the end is bounded by ∫ h, for any bound |H_q(t, x)| ≤ h(t), whatever the length of that part.
The perturbation vanishes off the three ends
(TauCeti.Floer.exists_cylindricalEndLift_of_pairOfPantsPerturbation_ne_zero), so the collars of
the three ends carry all of its curvature.
The components K_s and K_t of the perturbation in end coordinates are
TauCeti.Floer.pairOfPantsEndComponent H q 1 and TauCeti.Floer.pairOfPantsEndComponent H q I.
Main results #
TauCeti.SmoothTwoForm.hamiltonianCurvature_pairOfPantsEndComponent: the curvature on the end is±β'(±s - 1) H_q(t), for a manifold target; the vector-space case isTauCeti.SymplecticForm.hamiltonianCurvature_pairOfPantsEndComponent.TauCeti.SymplecticForm.hamiltonianCurvature_pairOfPantsEndComponent_eq_zero: it vanishes outside the collar1 ≤ ±s ≤ 2.TauCeti.SymplecticForm.integral_abs_hamiltonianCurvature_pairOfPantsEndComponent_le: the integrated curvature along any map over a rectangle[a₁, b₁] × [a₂, b₂]in the end is at most∫_{a₂}^{b₂} h, uniformly ina₁andb₁.TauCeti.SymplecticForm.abs_integral_floerEnergyDensity_pairOfPantsEndComponent_sub_le: consequently the Floer energy over such a rectangle differs from the symplectic area plus the Hamiltonian boundary terms∫ β(±s - 1) H_q(t, u(s, t)) dtat its two ends by at most∫ h.
References #
- M. Schwarz, Cohomology operations from S¹-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995), on perturbation data for the pair-of-pants product and its energy estimate.
- D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, Section 8.1, on the curvature of Hamiltonian perturbations and the resulting energy bounds.
The curvature on an end, for a manifold target. On the part ±s > 0 of the end q, the
curvature of the pair-of-pants perturbation in end coordinates is ±β'(±s - 1) H_q(t).
The curvature on an end. On the part ±s > 0 of the end q, the curvature of the
pair-of-pants perturbation in end coordinates is ±β'(±s - 1) H_q(t).
The curvature is supported in the collar. On the parts 0 < ±s < 1 and ±s > 2 of the
end q, the curvature of the pair-of-pants perturbation in end coordinates vanishes.
Uniform curvature bound on an end. Let h bound the Hamiltonian H_q of the end q on
[a₂, b₂], uniformly on the target. Over any rectangle [a₁, b₁] × [a₂, b₂] in the part
±s > 0 of the end, the integrated curvature of the pair-of-pants perturbation along any map u
is at most ∫_{a₂}^{b₂} h, independently of a₁ and b₁.
Uniform energy estimate on an end. Let H_q be C¹ and let h bound it on [a₂, b₂],
uniformly on the target. For a map u which is C¹ near a rectangle [a₁, b₁] × [a₂, b₂] in the
part ±s > 0 of the end q, the Floer energy of u over the rectangle for the pair-of-pants
perturbation differs from its symplectic area plus the Hamiltonian boundary terms
∫ β(±s - 1) H_q(t, u(s, t)) dt at s = b₁ and s = a₁ by at most ∫_{a₂}^{b₂} h,
independently of a₁ and b₁.