The energy identity on the pair of pants #
Let K be a C¹ Hamiltonian perturbation datum on the pair of pants S = ℂ ∖ {0, 1} which is
H_q(t) dt beyond some height T₀ on each of its three ends, for continuous period-one
Hamiltonians H_q, and let u be a C² map from S to a symplectic vector space (V, ω).
Assume that on each end q the actions A_{H_q} of the loops t ↦ u(φ_q(s, t))
(TauCeti.SymplecticForm.hamiltonianAction) converge to a limit A_q as the signed longitudinal
coordinate ±s tends to infinity. This file proves the energy identity of the pair-of-pants
product,
E(u) = A₀ + A₁ - A_∞ - ∫_S R_K(u),
where E(u) is the integral of the Floer energy density and R_K is the curvature of K. It
holds when the energy density and the curvature term are integrable, and also when the energy
density is merely nonnegative, as it is for a solution of the Floer equation, and the curvature
term is integrable; then the energy is finite. The two positive ends at 0 and 1 are the inputs
and the negative end at ∞ is the output. The actions cap each loop by the cone over it, which
fixes the area convention. Two caps of a loop differ by a sphere, and every sphere in V has area
zero because the symplectic form of a vector space is exact, ω = dλ. So the actions do not
depend on the caps, and the spherical correction ω(A) of a product coefficient vanishes.
Below the height T₀, K is only required to be C¹; beyond T₀ it must equal H_q(t) dt.
The identity therefore covers the pair-of-pants perturbation
TauCeti.Floer.pairOfPantsPerturbation, the family obtained from it by inserting continuation
necks on the ends, and its sums with compactly supported variations of the Hamiltonian term in the
core of S, such as the variations used to achieve transversality.
Main results #
SymplecticForm.integral_floerEnergyDensity_eq_sum_hamiltonianAction_of_eq_smul_endAngularForm: the energy identity for an integrable energy density, in terms of the limits of the end-loop actions.SymplecticForm.lintegral_floerEnergyDensity_eq_of_eq_smul_endAngularForm,SymplecticForm.integrable_floerEnergyDensity_of_eq_smul_endAngularForm: the energy identity for a nonnegative energy density, which is then integrable.SymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation_eq_sum_hamiltonianAction: the energy identity for the pair-of-pants perturbation.
References #
- 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.
- D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, Section 8.1, on the energy identity for Hamiltonian perturbations on punctured surfaces.
- P. Seidel, Fukaya categories and Picard–Lefschetz theory, Section 8, on perturbation data on surfaces with strip-like ends and their energy.
The energy identity for perturbations that are cylindrical on the ends. Let K be a
perturbation datum on S = ℂ ∖ {0, 1} which is C¹ jointly in the domain and target points, with
components K_s(z) = K(z)(1) and K_t(z) = K(z)(i) in the global coordinate z = s + i t, and
which is H_q(t) dt beyond the height T₀ on each end q, for continuous period-one
Hamiltonians H_q. Let u be a C² map from S to a symplectic vector space whose Floer energy
density and curvature term R_K(u) are integrable. Suppose that on each end q the actions of the
loops t ↦ u(φ_q(s, t)) tend to A_q as ±s → ∞. Then the energy of u is
E = A₀ + A₁ - A_∞ - ∫ R_K(u).
On the compact part of S below the height T₀ the datum is only required to be C¹: this
covers the pair-of-pants perturbation, its family with continuation necks, and its sums with
compactly supported variations.
The energy identity on the pair of pants. Let K be the pair-of-pants perturbation of
period-one C¹ Hamiltonians H_q on the 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, and let u be a
C² map from S to a symplectic vector space whose Floer energy density and curvature term
R_K(u) are integrable. Suppose that on each end q the actions of the loops
t ↦ u(φ_q(s, t)) tend to A_q as ±s → ∞. Then the energy of u is
E = A₀ + A₁ - A_∞ - ∫ R_K(u): the actions at the two positive (input) ends minus the action at
the negative (output) end, minus the integrated curvature.
The energy identity for a nonnegative energy density. Under the hypotheses of
integral_floerEnergyDensity_eq_sum_hamiltonianAction_of_eq_smul_endAngularForm, suppose that
the Floer energy density of u is nonnegative on S, as it is for a solution of the Floer
equation with tame almost complex structures. Then no integrability of the energy density is
needed: the energy of u, as a Lebesgue integral, is E = A₀ + A₁ - A_∞ - ∫ R_K(u), and in
particular it is finite.
A nonnegative energy density is integrable. Under the hypotheses of
lintegral_floerEnergyDensity_eq_of_eq_smul_endAngularForm, the Floer energy density of u is
integrable, so integral_floerEnergyDensity_eq_sum_hamiltonianAction_of_eq_smul_endAngularForm
applies to it.