The Floer energy identity for manifold targets #
For a map u : ℝ × ℝ → M and a Hamiltonian one-form K = Ks ds + Kt dt, the
perturbed derivative takes values in the actual tangent space T_(u z) M. Its symplectic
area density satisfies
ω(du - X_K) = ω(du) + d(K(u)) - R_K(u).
Here R_K = ∂s Kt - ∂t Ks - {Ks, Kt} uses the intrinsic Hamiltonian vector fields, with convention
ι_(X_H) ω = dH. The curvature of a cylindrical perturbation K = H_t dt, with no ds component
and independent of s, vanishes (TauCeti.SmoothTwoForm.hamiltonianCurvature_cylindrical). For a
nondegenerate two-form on a finite-dimensional model, the identity holds for differentiable maps and
jointly differentiable Hamiltonians on the domain times the manifold. Neither closedness of the
two-form nor the Floer equation is needed for this pointwise identity. For solutions of the equation
the density is nonnegative under tameness, and is half the standard quadratic energy density. The
almost complex structure may be chosen independently at each domain point, as is needed for
domain-dependent perturbations.
Integrating the identity over a rectangle [a₁, b₁] × [a₂, b₂] of the domain gives
∫ ω(du - X_K) = ∫ u^* ω + ∮ K(u) - ∫ R_K(u)
(SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity). The exterior derivative
d(K(u)) is a function on the domain alone, so the divergence theorem applies to it whatever
the target. The integrability of the energy density and of R_K(u) is assumed, and that of the
area density follows. If K(u) has compact support in the plane, the boundary term disappears
and the energy over the whole plane is ∫ u^* ω - ∫ R_K(u)
(SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_of_hasCompactSupport). This is the
form of the identity on a closed domain, such as a sphere whose perturbation data vanish near one
point. More generally, when the perturbation data themselves vanish outside a compact set and
u is only C¹ near that set, the energy over every region containing it is
∫ u^* ω - ∫ R_K(u)
(SmoothTwoForm.IsNondegenerate.setIntegral_floerEnergyDensity_of_forall_notMem_eq_zero).
Since d(K(u)) is linear in K, the energy plus the integrated curvature over a compact region
only depends on the data near its boundary: two data that agree outside the region give the same
value (IsNondegenerate.setIntegral_floerEnergyDensity_add_hamiltonianCurvature_eq_of_eqOn).
This compares a family of perturbations with a fixed datum that agrees with it far out on the
ends of a punctured domain.
A punctured domain carries no global coordinate, so these objects must also be compared across
changes of domain coordinates φ. For the pulled-back datum φ^*K
(TauCeti.Floer.perturbationPullback) and the reparametrized map u ∘ φ, the perturbed
derivative is (du - X_K) ∘ dφ, and both the energy density and the curvature are multiplied by
the Jacobian determinant det dφ. By the change of variables formula the energy and the
integrated curvature over a region may therefore be computed in any orientation-preserving
chart. These are the manifold-target versions of the identities in
TauCeti.Geometry.Symplectic.Floer.Pullback. The domain-derivative part of the curvature,
Green's theorem for K(u) and the change of variables formula do not see the target, and are
shared with them through
TauCeti.Floer.fderiv_perturbationPullback_sub_fderiv_perturbationPullback,
TauCeti.integral_fderiv_sub_fderiv_Icc and
TauCeti.integral_image_eq_integral_of_eq_abs_det_smul.
These are coordinate-domain energy identities for arbitrary manifold targets, not a compactness theorem or an identity involving caps at punctured ends. On a vector-space target with constant form they recover the vector-space definitions.
Main results #
SmoothTwoForm.hamiltonianCurvature_cylindrical: the curvature ofK = H_t dtvanishes.SmoothTwoForm.hamiltonianCurvature_congr_of_eventuallyEq,SmoothTwoForm.hamiltonianCurvature_continuation: the curvature is local in the domain, and a continuation perturbationK = β(s) H_t dthas curvatureβ'(s) H_t.SmoothTwoForm.hamiltonianCurvature_eq_zero_of_cylindrical: the curvature vanishes along a line near which the perturbation is cylindrical; the vector-space case isSymplecticForm.hamiltonianCurvature_eq_zero_of_cylindrical.SmoothTwoForm.IsNondegenerate.floerEnergyDensity_eq: the pointwise energy identity.SmoothTwoForm.IsNondegenerate.floerEnergyDensity_zero_left_nonneg,SmoothTwoForm.continuous_hamiltonianCurvature_zero_left_compandSmoothTwoForm.IsNondegenerate.continuous_floerEnergyDensity_zero_left: for a perturbationK = K_t dt, nonnegativity of the energy density on solutions, and continuity of the curvature and of the energy density.SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity: the energy identity on a rectangle.SmoothTwoForm.IsNondegenerate.integral_floerEnergyDensity_of_hasCompactSupport: the energy identity on the plane for a compactly supported perturbation.SmoothTwoForm.IsNondegenerate.setIntegral_floerEnergyDensity_of_forall_notMem_eq_zero: for perturbation data vanishing outside a compact set, the energy over any region containing that set is∫ u^* ω - ∫ R_K(u), with no boundary term; the vector-space case isSymplecticForm.setIntegral_floerEnergyDensity_of_forall_notMem_eq_zero.SmoothTwoForm.IsNondegenerate.integrableOn_floerEnergyDensity: over a compact set whereuandKareC¹, the energy density is integrable whenever the curvature is.IsNondegenerate.setIntegral_floerEnergyDensity_add_hamiltonianCurvature_eq_of_eqOn: changing the perturbation data inside a compact regionDdoes not change the energy plus the integrated curvature overD; the vector-space case isSymplecticForm.setIntegral_floerEnergyDensity_add_hamiltonianCurvature_eq_of_eqOn.SmoothTwoForm.perturbedMFDeriv_perturbationPullback,SmoothTwoForm.floerEnergyDensity_perturbationPullback,SmoothTwoForm.eval₂_mfderiv_comp_stdComplexLine,SmoothTwoForm.hamiltonianCurvature_perturbationPullback: naturality of the perturbed derivative, and the energy density, the area density and the curvature as two-forms on the domain.SmoothTwoForm.integral_floerEnergyDensity_image,SmoothTwoForm.integral_eval₂_mfderiv_stdComplexLine_image,SmoothTwoForm.integral_hamiltonianCurvature_image,SmoothTwoForm.lintegral_enorm_hamiltonianCurvature_image: invariance of the energy, the symplectic area, the integrated curvature and the curvature allowance under changes of domain coordinates.SmoothTwoForm.hamiltonianCurvature_comp_add_right,SmoothTwoForm.lintegral_enorm_hamiltonianCurvature_comp_add_right: translations of the domain translate the curvature and preserve the curvature allowance along a map.
References #
- D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, Section 8.1, on Hamiltonian perturbations, curvature, and the energy identity.
- P. Seidel, Fukaya categories and Picard–Lefschetz theory, Section 8.
The curvature ∂s Kt - ∂t Ks - {Ks, Kt} of a Hamiltonian one-form with manifold target.
Instances For
The defining curvature formula, separating domain derivatives and target Poisson bracket.
Without a ds component, the curvature of K = K_t dt is ∂_s K_t.
A cylindrical perturbation K = H_t dt, independent of s, has zero curvature.
The intrinsic perturbed derivative du - X_K, valued in T_(u z) M.
As with mfderiv, using this as an actual differential requires differentiability of u.
Instances For
Evaluation of the perturbed derivative on a domain tangent vector.
The real-coordinate component is ∂s u - X_(Ks).
The imaginary-coordinate component is ∂t u - X_(Kt).
The perturbed derivative is complex linear exactly when the coordinate Floer equation
∂s u - X_(Ks) + J (∂t u - X_(Kt)) = 0 holds in the target tangent fiber.
The symplectic area density of du - X_K for a manifold target.
Instances For
Floer energy density is the two-form evaluated on the two perturbed coordinate derivatives.
For the zero perturbation the Floer energy density is the symplectic area density
ω(∂_s u, ∂_t u).
The pointwise energy identity for a manifold target: symplectic area plus the exterior
derivative of K(u), minus Hamiltonian curvature.
On a solution the density is half the standard quadratic energy density of the perturbed derivative. The almost complex structure here is a structure on a single tangent fiber.
A Floer solution has nonnegative energy density for a tamed structure on the target fiber.
A solution of the Floer equation ∂_s u + J (∂_t u - X_{K_t}(u)) = 0 for a perturbation
K = K_t dt without ds component has nonnegative energy density at z when J is tamed by ω
at u z.
For a C¹ map u and a C¹ family of Hamiltonians K_t, the curvature ∂_s K_t of the
perturbation K = K_t dt, evaluated along u, is continuous.
For a C¹ map u and a C¹ family of Hamiltonians K_t, the Floer energy density of the
perturbation K = K_t dt is continuous.
Local formulas for the curvature #
The curvature only depends on the germs of the perturbation data at the domain point.
Translating the perturbation data by v translates the curvature by v.
If both components of a Hamiltonian one-form vanish near a domain point, its curvature vanishes there. No differentiability assumption is needed.
If both components of a Hamiltonian one-form vanish off a closed domain set, its curvature also vanishes off that set. No differentiability assumption is needed.
Adding a perturbation supported in a closed set inside an open region where the original one-form vanishes adds their curvatures exactly: there are no mixed Poisson-bracket terms.
The zero perturbation has zero curvature.
A continuation perturbation K = β(s) H_t dt has curvature β'(s) H_t.
A datum K = K_s ds + K_t dt with K_s = 0 and K_t independent of s near the line
s = c has zero curvature along that line.
Changes of domain coordinates #
The Hamiltonian vector field of a component of the pullback φ^*K is the corresponding
combination of the Hamiltonian vector fields of K_s and K_t at φ w.
Naturality of the perturbed derivative. The perturbed derivative of the reparametrized
map u ∘ φ for the pulled-back datum φ^*K is (du - X_K) ∘ dφ, in the tangent space
T_(u (φ w)) M.
The energy density is a two-form. The Floer energy density of u ∘ φ for the
pulled-back datum φ^*K is the Jacobian determinant of φ times the energy density of u for
K at φ w. No nondegeneracy of the two-form is needed.
The area density is a two-form. The symplectic area density ω(∂_s (u ∘ φ), ∂_t (u ∘ φ))
at w is the Jacobian determinant of φ times the area density ω(∂_s u, ∂_t u) at φ w.
The curvature is a two-form. For a C² change of coordinates φ, the curvature of the
pulled-back datum φ^*K is the Jacobian determinant of φ times the curvature of K at φ w.
Invariance of the energy under changes of coordinates. For a change of coordinates φ
that is injective and orientation preserving on a measurable set s, the energy of u for K
over φ '' s is the energy of u ∘ φ for φ^*K over s.
Invariance of the symplectic area under changes of coordinates. For a change of
coordinates φ that is injective and orientation preserving on a measurable set s, the
symplectic area ∫ ω(∂_s u, ∂_t u) of u over φ '' s is the symplectic area of u ∘ φ
over s.
Invariance of the integrated curvature under changes of coordinates. For a C² change of
coordinates φ that is injective and orientation preserving on a measurable set s, the integral
of the curvature R_K(u) over φ '' s is the integral of R_{φ^*K}(u ∘ φ) over s.
Invariance of the curvature allowance under changes of coordinates. For a C² change of
coordinates φ that is injective on a measurable set s, the lower Lebesgue integral of
|R_K(u)| over φ '' s is that of |R_{φ^*K}(u ∘ φ)| over s. No orientation, integrability
or measurability hypothesis is needed, so curvature allowances along arbitrary maps u may be
computed in any chart.
Translations of the domain preserve the integrated curvature along a map.
The energy identity on a rectangle, for a manifold target. Let u and the perturbation
data K = K_s ds + K_t dt be C¹ over an open neighbourhood U of a rectangle
[a₁, b₁] × [a₂, b₂], for a nondegenerate two-form on a finite-dimensional manifold. If the
energy density and the curvature R_K(u) are integrable over the rectangle, then the energy over
the rectangle is the symplectic area ∫ u^* ω, plus the integral of the pulled-back one-form
K(u) = K_s(u) ds + K_t(u) dt over the positively oriented boundary, minus the integral of
R_K(u). Closedness of the two-form is not used.
The energy identity on the plane, for a compactly supported perturbation. Let u and the
perturbation data K = K_s ds + K_t dt be C¹ on the whole plane, for a nondegenerate two-form
on a finite-dimensional manifold, and suppose that the pulled-back one-form
K(u) = K_s(u) ds + K_t(u) dt has compact support. If the energy density and the curvature
R_K(u) are integrable over the plane, then the energy is the symplectic area ∫ u^* ω minus the
integral of R_K(u): the exterior derivative d(K(u)) integrates to zero. This applies to
perturbation data on a closed surface that vanish near a point, written in the plane chart of the
complement of that point.
The energy identity for compactly supported perturbation data, for a manifold target.
Let the perturbation data K = K_s ds + K_t dt vanish outside a compact set C of the domain,
and let u and K be C¹ over an open neighbourhood U of C, for a nondegenerate two-form on
a finite-dimensional manifold. Then for every measurable set D ⊇ C over which the energy density
and the curvature R_K(u) are integrable, the energy over D is the symplectic area ∫_D u^* ω
minus the integral of R_K(u): the exterior derivative d(K(u)) is compactly supported, so the
boundary term of the energy identity vanishes. Closedness of the two-form is not used.
Integrability of the energy density, for a manifold target. Let u and the perturbation
datum K = K_s ds + K_t dt be C¹ over an open set U of the domain, for a nondegenerate
two-form on a finite-dimensional manifold, and let T ⊆ U be compact. If the curvature R_K(u) is
integrable over T, so is the Floer energy density: by the pointwise energy identity it differs
from R_K(u) by the symplectic area density and the exterior derivative d(K(u)), both continuous
on U.
Changing the perturbation inside a compact region, for a manifold target. Let u and two
perturbation data K = K_s ds + K_t dt and K' = K'_s ds + K'_t dt be C¹ over an open set U
of the domain, for a nondegenerate two-form on a finite-dimensional manifold, and let D ⊆ U be
compact. If K and K' agree at every point of U outside D, and the energy densities and the
curvatures along u are integrable over D, then the energy plus the integrated curvature over
D is the same for both data:
∫_D E_K(u) + ∫_D R_K(u) = ∫_D E_{K'}(u) + ∫_D R_{K'}(u).
By the pointwise energy identity the difference of the two sides is the integral of the exterior
derivative d((K - K')(u)), and (K - K')(u) is supported in D. In particular the boundary
term of the energy identity over D only sees the datum near the boundary of D.
A constant smooth two-form has the same Hamiltonian curvature as the vector-space form.
Under the canonical target tangent identification the intrinsic perturbed derivative for a constant form is the vector-space perturbed derivative.
The intrinsic energy density for a constant form agrees with the vector-space density.
For a constant form, the intrinsic symplectic area density of a map of the plane is the
vector-space area density ω(∂_s u, ∂_t u).
A datum K = K_s ds + K_t dt on a symplectic vector space with K_s = 0 and K_t independent
of s near the line s = c has zero curvature along that line. This is the constant-form case of
TauCeti.SmoothTwoForm.hamiltonianCurvature_eq_zero_of_cylindrical.
The energy identity for compactly supported perturbation data. Let the perturbation data
K = K_s ds + K_t dt vanish outside a compact set C of the domain, and let u and K be C¹
over an open neighbourhood U of C. Then for every measurable set D ⊇ C over which the energy
density is integrable, the energy over D is the symplectic area ∫_D u^* ω minus the integral of
the curvature R_K(u). This is the constant-form case of
TauCeti.SmoothTwoForm.IsNondegenerate.setIntegral_floerEnergyDensity_of_forall_notMem_eq_zero; the
curvature is continuous here, so its integrability is automatic.
Changing the perturbation inside a compact region. Let u and two perturbation data K
and K' be C¹ over an open set U of the domain, and let D ⊆ U be compact. If K and K'
agree at every point of U outside D, then the energy plus the integrated curvature over D is
the same for both data:
∫_D E_K(u) + ∫_D R_K(u) = ∫_D E_{K'}(u) + ∫_D R_{K'}(u).
This is the constant-form case of
SmoothTwoForm.IsNondegenerate.setIntegral_floerEnergyDensity_add_hamiltonianCurvature_eq_of_eqOn;
all four integrands are continuous on U, so their integrability over D is automatic.