Logarithmic coordinates on the cylindrical ends of the thrice-punctured sphere #
Each end lift cylindricalEndLift q of the thrice-punctured sphere is a scaled exponential
z ↦ c_q + b_q exp (-2πz) (Puncture.cylindricalEndLift_eq), with (c_q, b_q) equal to
(0, 1/2), (1, -1/2) and (0, 2) at 0, 1 and ∞. A logarithm inverts it: the end
coordinate ζ_q(w) = -log ((w - c_q) / b_q) / (2π) satisfies cylindricalEndLift q (ζ_q w) = w
for w ≠ c_q (Puncture.cylindricalEndLift_endCoord), and recovers a cylinder point z up to
an integer multiple of i (Puncture.exists_endCoord_cylindricalEndLift), so the end lift is
injective on every strip t₀ ≤ Im z < t₀ + 1 (Puncture.injOn_cylindricalEndLift). Its real part
is the longitudinal coordinate s, a smooth function away from c_q
(Puncture.contDiffAt_re_endCoord). Its imaginary part is the angular coordinate t, with values
in [-1/2, 1/2) (Puncture.im_endCoord_mem_Ico); it jumps by 1 across a ray, so only
period-one functions of it are smooth (Puncture.contMDiffAt_comp_im_endCoord). The angular
one-form dt of the end is v ↦ Im (v / (-2π (w - c_q))) (Puncture.endAngularForm), and it
pulls back to dt along the end lift (Puncture.endAngularForm_cylindricalEndLift).
The ends at 0 and 1 are positive, parameterized by s > 0, and the end at ∞ is negative,
parameterized by s < 0; Puncture.endSign records the sign. Points with positive signed
longitudinal coordinate lie in the standard punctured neighborhood of the puncture
(Puncture.exists_mem_neighborhood_cylindricalEndLift), hence in the thrice-punctured sphere
(Puncture.cylindricalEndLift_ne_zero_and_ne_one), and the three ends have disjoint images
(Puncture.cylindricalEndLift_ne_cylindricalEndLift), so a point has positive signed longitudinal
coordinate on at most one end (Puncture.endSign_mul_re_endCoord_nonpos). Bounding all three signed
longitudinal coordinates by R cuts out the truncation S_R of the thrice-punctured sphere
(TauCeti.ThricePuncturedSphere.truncation), a compact set
(TauCeti.ThricePuncturedSphere.isCompact_truncation) inside S
(TauCeti.ThricePuncturedSphere.truncation_subset) over which integrals of data on S are
taken. In the real coordinates ℝ × ℝ ≃ ℂ the longitudinal coordinates are measurable
(Puncture.measurable_re_endCoord), and so are the truncation and its parts ±s ≤ c on an end
(TauCeti.ThricePuncturedSphere.measurableSet_preimage_truncation,
TauCeti.ThricePuncturedSphere.measurableSet_preimage_truncation_inter). The truncations
increase with R and exhaust S (TauCeti.ThricePuncturedSphere.truncation_mono,
TauCeti.ThricePuncturedSphere.iUnion_truncation), so integrals over S are their limits as
R → ∞.
These coordinates are what is needed to install data prescribed on the ends, such as the
Hamiltonian terms H(t) dt of Floer perturbation data, as smooth data on the punctured sphere.
See Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Symplectic Topology, Chapter 4,
for the cylindrical end coordinates of a punctured Riemann surface.
The sign of an end: 1 at the positive ends at 0 and 1, parameterized by s > 0, and
-1 at the negative end at ∞, parameterized by s < 0.
Instances For
The sign of an end is nonzero.
Bounds |H_q| ≤ B_q give the one-sided bounds ±h_q ≤ ±H_q for the constants
h_q = ∓B_q.
The center of an end lift: cylindricalEndLift q z - q.endCenter is a multiple of
exp (-2πz). It is 0 at the ends at 0 and ∞, and 1 at the end at 1.
Instances For
The end lifts are nonconstant exponentials.
Every end lift is a scaled exponential about its center.
In polar form, the end lift traverses the circle of radius ‖b_q‖ exp (-2πs) about the
center c_q of the end clockwise, starting at the argument of the scale b_q.
The derivative of an end lift at z is -2π times the displacement of its value from the
center of the end.
An end lift never takes the value of the center of its end.
The logarithmic end coordinate ζ_q(w) = -log ((w - c_q) / b_q) / (2π). It inverts the end
lift (cylindricalEndLift_endCoord); its real part is the longitudinal coordinate and its
imaginary part an angular coordinate, determined by the point only modulo 1.
Instances For
Approaching the center of an end, its longitudinal coordinate tends to ∞.
On the closed disc ‖w‖ ≤ 2 exp (2πR) with R ≥ 0, the signed longitudinal coordinate -s of
the end at ∞ is at most R: the disc is bounded by the circle s = -R of that end.
On the punctured disc 0 < ‖w - c_q‖ ≤ exp (-2πr) / 2 about the center of a positive end q,
the longitudinal coordinate of that end is at least r: the disc is bounded by the circle s = r
of the end.
Off the open disc of radius 1/2 about the center of a positive end q, the longitudinal
coordinate of that end is nonpositive: the disc is bounded by the circle s = 0 of the end.
The end coordinate is a right inverse of the end lift away from the center of the end.
The end coordinate recovers a cylinder point from its image under the end lift, up to an integer shift of the angular coordinate.
The end coordinate recovers the longitudinal coordinate of a cylinder point from its image under the end lift.
An end lift is injective on every horizontal strip t₀ ≤ Im z < t₀ + 1 of height one: two
points with the same image differ by an integer multiple of i.
A smooth family f t x of period one in t, evaluated at the angular coordinate of an end,
is smooth away from the center of the end, jointly with the parameter x in a manifold. The
angular coordinate itself jumps by 1 across a ray; periodicity removes the jump.
The angular form pulls back to dt along the end lift: it sends the image φ'(z) v of a
tangent vector v of the cylinder to Im v.
The angular form is smooth away from the center of the end.
A point of an end with positive signed longitudinal coordinate lies in the standard punctured neighborhood of the puncture.
The longitudinal coordinate of an end, in the real coordinates ℝ × ℝ ≃ ℂ, is measurable.
The truncation S_R of the thrice-punctured sphere S = ℂ ∖ {0, 1}: the points of S at
which the signed longitudinal coordinate ±s of every end is at most R. It is S with the
parts ±s > R of its three cylindrical ends removed, and it is compact
(TauCeti.ThricePuncturedSphere.isCompact_truncation).
Instances For
For R > 0, the boundary circle ±s = R of each end lies in the truncation S_R.
The truncation is cut out by norm inequalities: it is the closed annulus
exp (-2πR) / 2 ≤ ‖w‖ ≤ 2 exp (2πR) with the open disc of radius exp (-2πR) / 2 about 1
removed.
The points of the disc ‖w‖ ≤ 2 exp (2πR) outside the truncation S_R lie in the open disc
‖w‖ < 2 exp (2π): for R ≥ 0 they are within exp (-2πR) / 2 ≤ 1 / 2 of 0 or 1, and for
R < 0 the disc is already smaller.
The truncation S_R lies in the closed annulus exp (-2πR) / 2 ≤ ‖w‖ ≤ 2 exp (2πR).
The points of the closed annulus exp (-2πR) / 2 ≤ ‖w‖ ≤ 2 exp (2πR) outside the truncation
S_R lie in the open disc of radius exp (-2πR) / 2 about 1.
The truncation of the thrice-punctured sphere is compact.
The truncation S_R of the thrice-punctured sphere is measurable in the real coordinates.
The part ±s ≤ c of the end q of the truncation S_R is measurable in the real
coordinates.
The truncations of the thrice-punctured sphere increase with the cutoff.
The poles 0 and 1 lie outside every truncation.