The energy of a pair of pants in terms of the actions of its ends #
Let K be the pair-of-pants perturbation TauCeti.Floer.pairOfPantsPerturbation of period-one
Hamiltonians H_q on the three ends of S = ℂ ∖ {0, 1}, with values in a symplectic vector space
(V, ω), and let u be a map which is C² on S. For a truncation S_R of S
(TauCeti.ThricePuncturedSphere.truncation), write ℓ_q(R) for the loop
t ↦ u(φ_q(±R + i t)) along the boundary circle ±s = R of the end q, where φ_q is the end
lift Puncture.cylindricalEndLift. This file rewrites the energy identity of
TauCeti.Geometry.Symplectic.Floer.PairOfPants.Curvature with Hamiltonian actions
A_H(x) = a(x) + ∫₀¹ H(t, x(t)) dt (TauCeti.SymplecticForm.hamiltonianAction):
∫_{S_R} E = A_{H_0}(ℓ_0(R)) + A_{H_1}(ℓ_1(R)) - A_{H_∞}(ℓ_∞(R)) - ∫_S R_K(u)
for R ≥ 2
(integral_floerEnergyDensity_pairOfPantsPerturbation_truncation_eq_sum_hamiltonianAction).
The symplectic area over S_R becomes the signed sum of the loop areas of the boundary circles by
Stokes' formula TauCeti.SymplecticForm.integral_fderiv_apply_truncation, and the curvature
integral over S_R is the curvature integral over all of S, because the curvature vanishes on
the cylindrical parts ±s > 2 of the ends
(SymplecticForm.hamiltonianCurvature_pairOfPantsPerturbation_eq_zero_of_two_lt); no bound on
the Hamiltonians is needed for this. If |H_q| ≤ B_q, then |∫_S R_K| ≤ B_0 + B_1 + B_∞, so
the energy over S_R differs from the signed sum of the actions by at most B_0 + B_1 + B_∞,
uniformly in R
(SymplecticForm.abs_integral_floerEnergyDensity_pairOfPantsPerturbation_truncation_sub_sum_le).
For a solution of the Floer equation on S_R with tame almost complex structures the energy over
S_R is nonnegative, which gives the action bound
A_{H_∞}(ℓ_∞(R)) ≤ A_{H_0}(ℓ_0(R)) + A_{H_1}(ℓ_1(R)) + B_0 + B_1 + B_∞ for every R ≥ 2
(SymplecticForm.hamiltonianAction_le_add_add_sum).
For a solution of the Floer equation (du - X_K)^{0,1} = 0 with tame almost complex structures
the energy density is nonnegative. If the actions of the three end loops converge to limits a_q
as R → ∞, then the energy of the solution over all of S is
E(u) = a_0 + a_1 - a_∞ - ∫_S R_K(u)
(SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_eq, an instance of
SymplecticForm.lintegral_floerEnergyDensity_eq_of_eq_smul_endAngularForm). This is the energy
identity E = A₁ + A₂ - A₀ + ω(A) - ∫ R_K of the pair-of-pants product for a vector space
target, on which every sphere has zero area. If |H_q| ≤ B_q, the curvature bound gives the
energy bound E(u) ≤ a_0 + a_1 - a_∞ + B_0 + B_1 + B_∞
(SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_le) and the action bound
a_∞ ≤ a_0 + a_1 + B_0 + B_1 + B_∞ for the output, the limit of the action bounds on the
truncations (SymplecticForm.le_add_add_sum_of_tendsto_hamiltonianAction): the potential bound
under which the counts of such solutions form a supported product kernel
(TauCeti.NovikovModule.kernelProduct).
The constant B_0 + B_1 + B_∞ can be replaced by the one-sided constant
-∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞ for any functions of time, integrable over one period, with
h_q ≤ H_q on the inputs and H_∞ ≤ h_∞ on the output, and then no bound on |H_q| is needed
(SymplecticForm.sum_mul_integral_le_setIntegral_hamiltonianCurvature_pairOfPantsPerturbation).
This gives the action bounds
A_{H_∞}(ℓ_∞(R)) ≤ A_{H_0}(ℓ_0(R)) + A_{H_1}(ℓ_1(R)) - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞ on each
truncation (SymplecticForm.hamiltonianAction_le_add_add_sub_sum) and its limit
a_∞ ≤ a_0 + a_1 - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞
(SymplecticForm.le_add_add_sub_sum_of_tendsto_hamiltonianAction), with the corresponding energy
bounds. A change of normalization H_q ↦ H_q + c_q(t), h_q ↦ h_q + c_q(t) changes the actions and
the constant by the same amounts ∫₀¹ c_q, so these bounds are invariant under it.
When the pair of pants is a piece of a larger domain, glued along the end q, the map is only
defined on part of that end. Cutting S_R along the circle ±s = c of the end q, 2 ≤ c ≤ R,
the energy over the part ±s ≤ c of S_R of a map which is C² only on the part ±s < c + 1 of
S is
∑_{r ≠ q} ±A_{H_r}(ℓ_r(R)) + ±A_{H_q}(ℓ_q(c)) - ∫ R_K(u),
with the curvature integrated over the same part, by
SymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation_inter_eq_sum_hamiltonianAction.
It follows from the identity on S_R for u cut off beyond the circle ±s = c + 3/4: on the
outer part c < ±s ≤ R of the end q, where the perturbation is H_q(t) dt, the energy of the
cut-off map is the change of action from the circle ±s = c to the circle ±s = R, whatever
the cut-off map is there.
References #
- D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, Section 8.1, on the energy identity for Hamiltonian perturbations.
- M. Schwarz, Cohomology operations from S¹-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995), on the energy estimate for the pair-of-pants product.
For the constants h_q = ∓B_q, the one-sided constant ∑ q, ε_q ∫₀¹ h_q of the action
bounds is -(B_0 + B_1 + B_∞).
The curvature of the pair-of-pants perturbation vanishes on S outside the truncation S_R
for R ≥ 2, which contains the collars 1 ≤ ±s ≤ 2 of the ends. So along any map u, the
integrated curvature over S is the integrated curvature over S_R.
The energy identity on the truncated pair of pants, in terms of actions. Let K be the
pair-of-pants perturbation of period-one C¹ Hamiltonians H_q on the three ends of
S = ℂ ∖ {0, 1}, with components K_s(z) = K(z)(1) and K_t(z) = K(z)(i) in the global coordinate
z = s + i t of ℂ, and let u be a map which is C² on S. For R ≥ 2, the Floer energy of
u over the truncation S_R is the signed sum of the Hamiltonian actions of the loops
ℓ_q(t) = u(φ_q(±R + i t)) along its three boundary circles, minus the integrated curvature over
S:
∫_{S_R} E = ∑ q, ±A_{H_q}(ℓ_q) - ∫_S R_K(u),
with the sign + on the input ends 0 and 1 and - on the output end ∞.
The energy of a pair of pants. Let K be the pair-of-pants perturbation of period-one
C¹ Hamiltonians H_q on the three ends of S = ℂ ∖ {0, 1}, and let u be a map which is C²
on S and solves the Floer equation (du - X_K)^{0,1} = 0 there for almost complex structures
J(z) tamed by ω. If the actions of the loops ℓ_q(t) = u(φ_q(±R + i t)) along the circles
±s = R of the ends converge to a_q as R → ∞, then the energy of u is
E(u) = a_0 + a_1 - a_∞ - ∫_S R_K(u).
The energy and the actions on a truncated pair of pants. If the end Hamiltonians are
bounded, |H_q| ≤ B_q, then for every R ≥ 2 the Floer energy of u over the truncation S_R
differs from the signed sum A_{H_0}(ℓ_0) + A_{H_1}(ℓ_1) - A_{H_∞}(ℓ_∞) of the actions of its
boundary loops by at most B_0 + B_1 + B_∞, uniformly in R.
One-sided action bounds #
The one-sided energy bound on a truncated pair of pants. Let h_q be functions of time,
integrable over one period, with h_q ≤ H_q on the input ends and H_∞ ≤ h_∞ on the output
end, that is ±h_q(t) ≤ ±H_q(t, x). Then for every R ≥ 2 the Floer energy of u over the
truncation S_R is at most the signed sum of the actions of its boundary loops minus
∫₀¹ h_0 + ∫₀¹ h_1 - ∫₀¹ h_∞:
∫_{S_R} E ≤ A_{H_0}(ℓ_0) + A_{H_1}(ℓ_1) - A_{H_∞}(ℓ_∞) - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞.
The one-sided action bound on a truncated pair of pants. Let u be a map which is C²
on S = ℂ ∖ {0, 1} and solves the Floer equation (du - X_K)^{0,1} = 0 on the truncation S_R,
R ≥ 2, for almost complex structures J(z) tamed by ω, where K is the pair-of-pants
perturbation of period-one C¹ Hamiltonians H_q. If h_q are functions of time, integrable
over one period, with h_q ≤ H_q on the input ends and H_∞ ≤ h_∞ on the output end, then
A_{H_∞}(ℓ_∞) ≤ A_{H_0}(ℓ_0) + A_{H_1}(ℓ_1) - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞
for the loops ℓ_q(t) = u(φ_q(±R + i t)) on the boundary of S_R. The constant only depends on
lower bounds for the input Hamiltonians and an upper bound for the output Hamiltonian, and it
shifts by ∫₀¹ c_q exactly as the actions do under a change of normalization
H_q ↦ H_q + c_q(t).
The one-sided energy bound for a pair of pants. Under the hypotheses of
SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_eq, if h_q are
functions of time, integrable over one period, with h_q ≤ H_q on the input ends and
H_∞ ≤ h_∞ on the output end, then
E(u) ≤ a_0 + a_1 - a_∞ - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞.
The one-sided action bound for a pair of pants. Under the hypotheses of
SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_eq, if h_q are
functions of time, integrable over one period, with h_q ≤ H_q on the input ends and
H_∞ ≤ h_∞ on the output end, then the limiting actions of the ends satisfy
a_∞ ≤ a_0 + a_1 - ∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞.
This is the limit R → ∞ of SymplecticForm.hamiltonianAction_le_add_add_sub_sum.
Action bounds for bounded Hamiltonians #
If |H_q| ≤ B_q, the constants h_q = ∓B_q satisfy the one-sided bounds ±h_q ≤ ±H_q, and the
one-sided constant -∫₀¹ h_0 - ∫₀¹ h_1 + ∫₀¹ h_∞ becomes B_0 + B_1 + B_∞.
The energy bound for a pair of pants. Under the hypotheses of
SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_eq, if moreover
|H_q| ≤ B_q, then the energy of u is at most a_0 + a_1 - a_∞ + B_0 + B_1 + B_∞.
The action bound on a truncated pair of pants. Let K be the pair-of-pants perturbation
of period-one C¹ Hamiltonians H_q with |H_q| ≤ B_q, and let u be a map which is C² on
S = ℂ ∖ {0, 1} and solves the Floer equation (du - X_K)^{0,1} = 0 on the truncation S_R,
R ≥ 2, for almost complex structures J(z) tamed by ω. Then the action of the output loop
ℓ_∞(t) = u(φ_∞(-R + i t)) is at most the sum of the actions of the input loops
ℓ_q(t) = u(φ_q(R + i t)) plus B_0 + B_1 + B_∞:
A_{H_∞}(ℓ_∞) ≤ A_{H_0}(ℓ_0) + A_{H_1}(ℓ_1) + B_0 + B_1 + B_∞.
No convergence of the actions as R → ∞ is assumed.
The action bound for a pair of pants. Under the hypotheses of
SymplecticForm.lintegral_floerEnergyDensity_pairOfPantsPerturbation_eq, if moreover
|H_q| ≤ B_q, then the limiting action of the output end is at most the sum of the limiting
actions of the input ends plus B_0 + B_1 + B_∞:
a_∞ ≤ a_0 + a_1 + B_0 + B_1 + B_∞.
This is the limit R → ∞ of SymplecticForm.hamiltonianAction_le_add_add_sum.
Truncations with one end cut short #
The energy identity on a rectangle of an end. Let u be C² on S and let [a, b]
lie in the part ±s > 0 of the end q. For the cylindrical datum h(t) dt of a C¹
Hamiltonian h of period one, the energy of u ∘ φ_q over the rectangle [a, b] × [0, 1] in the
end coordinates is the change of action A_h(u ∘ φ_q(b, ·)) - A_h(u ∘ φ_q(a, ·)).
The energy identity on a truncated pair of pants with one end cut short. Let K be the
pair-of-pants perturbation of period-one C¹ Hamiltonians H_r, and let 2 ≤ c ≤ R. Cut the
truncation S_R along the circle ±s = c of the end q, keeping the part ±s ≤ c. For a map u
which is C² on the part ±s < c + 1 of S, the energy over this region is
∑_{r ≠ q} ±A_{H_r}(ℓ_r(R)) + ±A_{H_q}(ℓ_q(c)) - ∫ R_K(u),
where ℓ_r(R)(t) = u(φ_r(±R + i t)) are the loops along the circles ±s = R of the other two
ends, ℓ_q(c)(t) = u(φ_q(±c + i t)) is the loop along the cut, and the curvature is integrated
over the same region. Nothing is assumed about u beyond the circle ±s = c + 1 of the end q,
which is how this applies to a pair of pants glued into a larger domain along that end.