Cylindrical ends of the thrice-punctured sphere #
The standard neighborhoods of 0, 1 and ∞ have actual period-one half-cylinder
parameterizations. The ends at 0 and 1 use s > 0; the end at infinity uses s < 0.
Their complex lifts are holomorphic with nonzero derivative. Thus all three coordinates
preserve the complex orientation, with the same cylinder orientation ds ∧ dt.
The three end images are disjoint, and their complement is a compact core.
These are the standard exponential end coordinates on a punctured Riemann surface;
see Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Symplectic Topology,
Chapter 4. The topology reuses the puncture charts and Mathlib's polar and additive-circle
homeomorphisms through TauCeti.Complex.UnitDisc.puncturedCylinder.
Radial cylindrical parameterization of any puncture neighborhood.
At the two finite punctures this is the positive end coordinate. At infinity the standard
negative end is obtained by reversing both cylinder coordinates, as in negativeEnd.
Instances For
In puncture coordinates the radial end is the standard punctured-disc cylinder.
The affine coordinate of an end is obtained by applying the inverse puncture chart.
The negative end at infinity, with s < 0 and period-one angular coordinate.
Both signs reverse relative to the positive radial coordinate, so its complex lift is
z ↦ 2 / exp (2πz) and remains holomorphic.
Instances For
The complex lifts of the two positive ends and the negative end at infinity.
The lift at infinity is also 2 / exp (2πz).
Instances For
The positive end at zero has exponential affine lift.
The positive end at one has exponential affine lift.
The negative end at infinity has exponential affine lift.
The negative lift is the inverse infinity chart applied to the positive exponential.
The end lifts at 0 and at ∞ are both centred at 0, so they differ by a translation of the
cylinder: the end lift at 0 is the end lift at ∞ shifted by log 4 / (2π).
On real angular lifts, the negative cylindrical end has its holomorphic lift.
Each lifted end map has the explicit nonzero complex derivative of its exponential.
All three lifted ends are holomorphic.
The lifted end maps are smooth at every order, also beyond their half-plane domains.
Every lifted end map is periodic with period one in its real angular coordinate.
The holomorphic lifts have no critical points, including at the chosen angular marker.
Each parameterization fills precisely its standard puncture neighborhood.
The negative end fills precisely the standard infinity neighborhood.
End images at distinct punctures are disjoint.
The complement of the two positive end images and the negative end image is compact.