The integrated curvature and the energy identity of the pair-of-pants perturbation #
The energy identity E = A₁ + A₂ - A₀ + ω(A) - ∫ R_K for pairs of pants turns a bound
∫_S sup_X |R_K| ≤ C on the curvature of the perturbation datum K into the lower bound on the
valuation of product coefficients and the upper bound on the energy of the curves they count.
This file computes such a constant C for the perturbation
TauCeti.Floer.pairOfPantsPerturbation built from period-one Hamiltonians H_q on the three ends
of the thrice-punctured sphere S = ℂ ∖ {0, 1}. The curvature bound holds for Hamiltonians on any
manifold M with a two-form ω, and the energy identity below for a finite-dimensional manifold
with a nondegenerate two-form, such as a closed symplectic manifold. Both specialize to a symplectic
vector space target V through the constant form SymplecticForm.constSmooth.
In the coordinates (s, t) of the end q the perturbation is β(±s) H_q(t) dt with the cutoff
β(±s) = smoothTransition (±s - 1)
(TauCeti.Floer.pairOfPantsPerturbation_cylindricalEndLift_of_pos), so its curvature there is the
continuation curvature ±β'(±s - 1) H_q(t)
(SmoothTwoForm.hamiltonianCurvature_pairOfPantsEndComponent), and the perturbation vanishes
off the parts ±s > 1 of the ends. Since the curvature is a two-form
(SmoothTwoForm.hamiltonianCurvature_perturbationPullback), its absolute integral over
S is the sum over the three ends of the integrals of β'(±s - 1) |H_q(t)| over half-cylinders,
and the cutoff has total increase ∫ β' = 1. Hence, if |H_q| ≤ B_q, then
∫_S |R_K(z, u(z))| dz ≤ B₀ + B₁ + B_∞
for every map u : S → M, measurable or not
(SmoothTwoForm.lintegral_enorm_hamiltonianCurvature_pairOfPantsPerturbation_le). The bound uses
only that K_s ds + K_t dt is a multiple of H_q(t) dt on each end, so the Poisson bracket term
of the curvature vanishes there, whatever the target; for a vector space target it specializes to
SymplecticForm.lintegral_enorm_hamiltonianCurvature_pairOfPantsPerturbation_le. The curvature
is computed in the global coordinate z = s + i t of S ⊆ ℂ, in which the ds and dt
components of K are its values on the tangent vectors 1 and i.
In the coordinates of an end, multiplied by the Jacobian of the end chart, the curvature along a
map u continuous on S is the continuous function ±β'(±s - 1) H_q(t, u), and it vanishes off
the images of the collars 1 ≤ ±s ≤ 2. So the curvature is integrable over S
(SmoothTwoForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation).
The same decomposition of S into its core, where K vanishes, and the three ends integrates
the energy identity over the truncation S_R of S
(TauCeti.ThricePuncturedSphere.truncation), where every ±s ≤ R. On each end the rectangle
identity TauCeti.SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity applies in cylinder
coordinates; the boundary terms along the angular sides cancel by periodicity, and those on the
circles ±s = 1 vanish with the cutoff. For a map u which is C¹ on S and R ≥ 1 this
gives
∫_{S_R} E = ∫_{S_R} u^*ω + β(R - 1) ∑ q, ±∫₀¹ H_q(t, u(φ_q(±R + i t))) dt - ∫_{S_R} R_K(u),
with the sign + on the input ends 0, 1 and - on the output end ∞. Here the area density
ω(∂_s u, ∂_t u) is continuous on S, so integrable over the compact set S_R, and so is the
energy density, since it differs from the area density by the curvature and the exterior
derivative d(K(u)), which is continuous on S. With the curvature bound, the energy over S_R
therefore differs from the area plus the Hamiltonian boundary terms by at most B₀ + B₁ + B_∞,
uniformly in R. Letting R → ∞
gives the energy identity on the whole pair of pants, in
TauCeti.Geometry.Symplectic.Floer.PairOfPants.Energy.Asymptotic.
The two-sided constant B₀ + B₁ + B_∞ does not behave well under a change of normalization
H_q ↦ H_q + c_q(t), which leaves the Floer equation unchanged and shifts the actions of the ends
by ∫₀¹ c_q. A one-sided constant transforms correctly. 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.
In the coordinates of the end q the curvature, multiplied by the Jacobian of the end chart, is
±β'(±s - 1) H_q(t), and β' ≥ 0 has integral one. Since the curvature is supported in the
disjoint images of the strips of the ends, the change of variables formula gives
∫₀¹ h_0 + ∫₀¹ h_1 - ∫₀¹ h_∞ ≤ ∫_S R_K(u).
For the best choices h_q = inf H_q(t, ·) on the inputs and h_∞ = sup H_∞(t, ·), when they are
integrable, this is the estimate of Schwarz.
Main results #
TauCeti.SmoothTwoForm.lintegral_enorm_hamiltonianCurvature_pairOfPantsPerturbation_le: the integrated curvature of the pair-of-pants perturbation along any map into a manifold is at most the sum of bounds for the three end Hamiltonians.SmoothTwoForm.abs_setIntegral_hamiltonianCurvature_pairOfPantsPerturbation_le: hence the integral of the curvature over any set has absolute value at most the same sum.SmoothTwoForm.hamiltonianCurvature_pairOfPantsPerturbation_eq_zero_of_two_lt: the curvature vanishes where the perturbation is cylindrical, beyond the collars1 ≤ ±s ≤ 2of the ends.SymplecticForm.lintegral_enorm_hamiltonianCurvature_pairOfPantsPerturbation_le,SymplecticForm.abs_setIntegral_hamiltonianCurvature_pairOfPantsPerturbation_le,SymplecticForm.hamiltonianCurvature_pairOfPantsPerturbation_eq_zero_of_two_lt: the same statements for a symplectic vector space target.SmoothTwoForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation,SymplecticForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation: the curvature along a map continuous onSis integrable overS.SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation_truncation: the energy identity on the truncated pair of pants, for a manifold target; the vector-space case isSymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation_truncation.SymplecticForm.abs_integral_floerEnergyDensity_pairOfPantsPerturbation_truncation_sub_le: the resulting energy bound, uniform in the truncation; the manifold version isIsNondegenerate.abs_integral_floerEnergyDensity_pairOfPantsPerturbation_truncation_sub_lein the namespaceSmoothTwoForm.SymplecticForm.sum_mul_integral_le_setIntegral_hamiltonianCurvature_pairOfPantsPerturbation: the one-sided curvature bound.
References #
- 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.
- P. Seidel, Fukaya categories and Picard–Lefschetz theory, Section 8, on perturbation data on surfaces with strip-like ends and their energy.
- M. Schwarz, Cohomology operations from S¹-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995), on the one-sided action estimate for the pair-of-pants product.
The pieces of a truncation of the pair of pants #
The integrated curvature of the pair-of-pants perturbation, for a manifold target. Let K
be the pair-of-pants perturbation of period-one C¹ Hamiltonians H_q on a manifold M with a
two-form ω, attached to the three ends of S = ℂ ∖ {0, 1}, with |H_q| ≤ B_q, and write its
components in the global coordinate z = s + i t of ℂ as K_s(z) = K(z)(1) and
K_t(z) = K(z)(i). Then for every map u into M, the integral of |R_K(z, u(z))| over the
plane is at most B₀ + B₁ + B_∞.
The integral of the curvature of the pair-of-pants perturbation along any map into a manifold,
over any set, has absolute value at most B₀ + B₁ + B_∞ when |H_q| ≤ B_q.
The curvature vanishes on the cylindrical parts of the ends, for a manifold target. At a
point of S where the signed longitudinal coordinate of some end is greater than 2, the
perturbation is H_q(t) dt in the coordinates of that end, and its curvature vanishes. Together
with the vanishing of the perturbation on the core, this confines the curvature to the collars
1 ≤ ±s ≤ 2 of the three ends, a compact part of S.
The energy identity on the truncated pair of pants #
The curvature is integrable over the pair of pants, for a manifold target. Along a map u
which is continuous on S, the curvature of the pair-of-pants perturbation of period-one C¹
Hamiltonians on a manifold is integrable over S. It vanishes off the images of the collars
1 ≤ ±s ≤ 2 of the three ends, and in the coordinates of an end, multiplied by the Jacobian of
the end chart, it is the continuous function ±β'(±s - 1) H_q(t, u).
The energy identity on the truncated pair of pants, for a manifold target. Let K be the
pair-of-pants perturbation of period-one C¹ Hamiltonians H_q on a finite-dimensional manifold
with a nondegenerate two-form, 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 C¹ on S. For R ≥ 1,
the Floer energy over the truncation S_R of S is the symplectic area, plus β(R - 1) times
the signed sum over the three ends of the integrals of H_q(t) along the boundary circles
±s = R, minus the integrated curvature:
∫_{S_R} E = ∫_{S_R} u^*ω + β(R - 1) ∑ q, ±∫₀¹ H_q(t, u(φ_q(±R + i t))) dt - ∫_{S_R} R_K(u).
Here φ_q is the end lift of q, the sign is + on the input ends 0 and 1 and - on the
output end ∞, and β = smoothTransition is the cutoff of the perturbation, so β(R - 1) = 1
for R ≥ 2. The energy, area and curvature densities are integrable over S_R automatically.
The energy bound on the truncated pair of pants, for a manifold target. If the end
Hamiltonians are bounded, |H_q| ≤ B_q, then for every R ≥ 1 the Floer energy over the
truncation S_R differs from the symplectic area plus the Hamiltonian
boundary terms by at most B₀ + B₁ + B_∞, uniformly in R.
The integrated curvature of the pair-of-pants perturbation. 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
|H_q| ≤ B_q, and write its components in the global coordinate z = s + i t of ℂ as
K_s(z) = K(z)(1) and K_t(z) = K(z)(i). Then for every map u into the target, the integral
of |R_K(z, u(z))| over the plane is at most B₀ + B₁ + B_∞. This is the vector-space case of
TauCeti.SmoothTwoForm.lintegral_enorm_hamiltonianCurvature_pairOfPantsPerturbation_le.
The integral of the curvature of the pair-of-pants perturbation along any map, over any set,
has absolute value at most B₀ + B₁ + B_∞ when |H_q| ≤ B_q.
The energy identity on the truncated pair of pants #
The curvature vanishes on the cylindrical parts of the ends. At a point of S where the
signed longitudinal coordinate of some end is greater than 2, the perturbation is H_q(t) dt in
the coordinates of that end, and its curvature vanishes. Together with
hamiltonianCurvature_pairOfPantsPerturbation_eq_zero on the core, this confines the curvature to
the collars 1 ≤ ±s ≤ 2 of the three ends, a compact part of S.
The Floer energy density, the symplectic area density and the curvature of the pair-of-pants
perturbation, along a map that is C¹ on S, are integrable over every truncation.
The energy identity on the truncated 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}, 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 C¹ on S. For R ≥ 1, over the truncation S_R of S the Floer
energy of u is its symplectic area, plus β(R - 1) times the signed sum over the three ends of
the integrals of H_q(t) along the boundary circles ±s = R, minus the integrated curvature:
∫_{S_R} E = ∫_{S_R} u^*ω + β(R - 1) ∑ q, ±∫₀¹ H_q(t, u(φ_q(±R + i t))) dt - ∫_{S_R} R_K(u).
Here φ_q is the end lift of q, the sign is + on the input ends 0 and 1 and - on the
output end ∞, and β = smoothTransition is the cutoff of the perturbation, so β(R - 1) = 1
for R ≥ 2. This is the vector-space case of
SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation_truncation.
The energy bound on the truncated pair of pants. If the end Hamiltonians are bounded,
|H_q| ≤ B_q, then for every R ≥ 1 the Floer energy over the truncation S_R differs from the
symplectic area plus the Hamiltonian boundary terms by at most B₀ + B₁ + B_∞, uniformly in R.
The curvature is integrable over the pair of pants. Along a map u which is continuous
on S, the curvature of the pair-of-pants perturbation of period-one C¹ Hamiltonians is
integrable over S. This is the vector-space case of
SmoothTwoForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation.
One-sided curvature bounds #
The one-sided curvature bound on the 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). Along every map u continuous on S, the integrated
curvature of the pair-of-pants perturbation of period-one C¹ Hamiltonians H_q satisfies
∫₀¹ h_0 + ∫₀¹ h_1 - ∫₀¹ h_∞ ≤ ∫_S R_K(u).
Unlike the two-sided bound |∫_S R_K(u)| ≤ B_0 + B_1 + B_∞, it needs no bound on |H_q|, and
the constant transforms correctly under a change of normalization H_q ↦ H_q + c_q(t), which
shifts the actions of the ends by ∫₀¹ c_q.