The energy identity on the whole pair of pants #
Let K be the pair-of-pants perturbation TauCeti.Floer.pairOfPantsPerturbation built from
bounded period-one C¹ Hamiltonians H_q on the three ends of S = ℂ ∖ {0, 1}, and let u be
a map which is C¹ on S, with values in a finite-dimensional manifold M carrying a
nondegenerate two-form ω, such as a closed symplectic manifold. On every truncation S_R of S
the energy identity
∫_{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)
holds; this is
SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation_truncation.
This file lets R → ∞. Suppose that the loops t ↦ u(φ_q(±R + i t)) along the circles of the end
q converge pointwise to loops x_q as R → ∞, and that the energy density and the area density
of u are integrable over S. The curvature is integrable over S
(SmoothTwoForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation), the truncations exhaust
S, and dominated convergence on the boundary circles gives
∫_S E = ∫_S u^*ω + ∑ q, ±∫₀¹ H_q(t, x_q(t)) dt - ∫_S R_K(u),
the identity E = ∫ u^*ω + ∫_∂ K(u) - ∫ R_K(u) with the boundary term evaluated at the asymptotic
orbits, with the sign + on the input ends 0, 1 and - on the output end ∞. Hence the
energy differs from this topological part by at most B₀ + B₁ + B_∞ for bounds |H_q| ≤ B_q.
For a solution of the Floer equation with tamed almost complex structures the energy is
nonnegative, so the topological part is at least -(B₀ + B₁ + B_∞). Once the area is expressed
through the actions of capped orbits and the product class, this becomes the lower bound on the
valuation of the output monomials of the pair-of-pants product.
The results are stated for a manifold target, and the vector-space statements are their
specializations to the constant form SymplecticForm.constSmooth.
Main results #
TauCeti.Floer.tendsto_setIntegral_truncation,TauCeti.Floer.tendsto_intervalIntegral_endLoop: integrals overSare limits of integrals over its truncations, and integrals of a Hamiltonian along the end loops converge to its integral along their pointwise limits.SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation: the energy identity on the whole pair of pants, for a map with asymptotic loops at the ends; the vector-space case isSymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation.SmoothTwoForm.IsNondegenerate.abs_integral_floerEnergyDensity_pairOfPantsPerturbation_sub_le,SymplecticForm.abs_integral_floerEnergyDensity_pairOfPantsPerturbation_sub_le: the resulting energy bound.SmoothTwoForm.IsNondegenerate.neg_sum_le_integral_add_sum_of_isComplexLinearMap,SymplecticForm.neg_sum_le_integral_add_sum_of_isComplexLinearMap: the lower bound on the topological energy of a Floer solution.
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.
- 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.
Integrals over S are the limits of the integrals over its truncations.
If the loops of u along the circles ±s = R of the end q converge pointwise to a loop x
as R → ∞, then the integrals of a bounded continuous Hamiltonian along them converge to its
integral along x.
The energy identity on the pair of pants, for a manifold target. Let K be the
pair-of-pants perturbation of bounded 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. Let u be a map which is C¹ on S, whose loops
t ↦ u(φ_q(±R + i t)) along the circles of the end q converge pointwise to loops x_q as
R → ∞, and whose Floer energy density and symplectic area density are integrable over S. Then
∫_S E = ∫_S u^*ω + ∑ q, ±∫₀¹ H_q(t, x_q(t)) dt - ∫_S R_K(u),
with the sign + on the input ends 0, 1 and - on the output end ∞. The middle term is the
boundary integral ∫_∂ K(u) at the asymptotic orbits, and the curvature term is integrable
(SmoothTwoForm.integrableOn_hamiltonianCurvature_pairOfPantsPerturbation).
The energy bound on the pair of pants, for a manifold target. Under the hypotheses of
SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation, the Floer
energy over S differs from its topological part, the symplectic area plus the Hamiltonian terms
at the asymptotic orbits, by at most B₀ + B₁ + B_∞.
The topological energy of a pair-of-pants solution is bounded below, for a manifold
target. Let u solve the Floer equation (du - X_K)^{0,1} = 0 on S for almost complex
structures J z on the tangent spaces T_{u(z)} M tamed by the two-form, under the hypotheses of
SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_pairOfPantsPerturbation. Since its
energy is nonnegative, its symplectic area plus the Hamiltonian terms at the asymptotic orbits is
at least -(B₀ + B₁ + B_∞).
The energy identity on the pair of pants. Let K be the pair-of-pants perturbation of
bounded 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. Let u
be a map which is C¹ on S, whose loops t ↦ u(φ_q(±R + i t)) along the circles of the end
q converge pointwise to loops x_q as R → ∞, and whose Floer energy density and symplectic
area density are integrable over S. Then
∫_S E = ∫_S u^*ω + ∑ q, ±∫₀¹ H_q(t, x_q(t)) dt - ∫_S R_K(u),
with the sign + on the input ends 0, 1 and - on the output end ∞. The middle term is the
boundary integral ∫_∂ K(u) at the asymptotic orbits, and the curvature term is integrable
(integrableOn_hamiltonianCurvature_pairOfPantsPerturbation).
The energy bound on the pair of pants. Under the hypotheses of
SymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation, the Floer energy over S
differs from its topological part, the symplectic area plus the Hamiltonian terms at the
asymptotic orbits, by at most B₀ + B₁ + B_∞.
The topological energy of a pair-of-pants solution is bounded below. Let u solve the
Floer equation (du - X_K)^{0,1} = 0 on S for almost complex structures J z tamed by ω,
under the hypotheses of SymplecticForm.integral_floerEnergyDensity_pairOfPantsPerturbation.
Since its energy is nonnegative, its symplectic area plus the Hamiltonian terms at the asymptotic
orbits is at least -(B₀ + B₁ + B_∞).