Local structure of trajectories

of a quadratic differential

A description of the behaviour of the trajectories of a quadratic differential on an oriented Riemann surface in a neighbourhood of any point of this surface. Let be an oriented Riemann surface and let be a quadratic differential on ; let be the set of all zeros and simple poles of and let be the set of all poles of of order . The trajectories of form a regular family of curves on . Under an extension of the concept of a regular family of curves this remains true on also. The behaviour of the trajectories in neighbourhoods of points of is significantly more complicated. A complete description of the local structure of trajectories is given below.

a) For any point there is a neighbourhood of on and a homeomorphic mapping of onto the disc () such that a maximal open arc of each trajectory in goes to a segment on which is constant. Consequently, through each point of there passes a trajectory of that is either an open arc or a Jordan curve on .

b) For any point of order ( if is a zero and if is a simple pole) there is a neighbourhood of on and a homeomorphic mapping of onto the disc such that a maximal arc of each trajectory in goes to an open arc on which is constant. There are trajectories with ends at and with limiting tangential directions that make equal angles with each other.

c) Let be a pole of order . If a certain trajectory has an end at , then it tends to along one of directions making equal angles . There is a neighbourhood of on with the following properties: 1) every trajectory that passes through some point of in each of the directions either tends to or leaves ; 2) there is a neighbourhood of contained in and such that every trajectory that passes through some point of tends to in at least one direction, remaining in ; 3) if some trajectory lies entirely in and therefore tends to in both directions, then the tangent to this trajectory as is approached in the corresponding direction tends to one of two adjacent limiting positions. The Jordan curve obtained by adjoining to this trajectory bounds a domain containing points of the angle formed by the two adjacent limiting tangents. The tangent to any trajectory that has points in common with tends to these adjacent limiting positions as is approached in the two directions. By means of a suitable branch of the function the domain is mapped onto the half-plane (where is a real number); and 4) for every pair of adjacent limiting positions there is a trajectory having the properties described in 3).

d) Let be a pole of order two and let be the local parameter in terms of which is the point . Suppose that has (for some choice of the branch of the root) the following expansion in a neighbourhood of :

where and are real constants and are complex constants. The structure of the images of the trajectories of the differential in the -plane is determined by which of the following three cases holds.

Case I: , . For sufficiently small the image of each trajectory that intersects the disc tends to in one direction, and leaves in the other direction. Both the modulus and the argument of vary monotonically on the image of the trajectory in . Each image of a trajectory twists around the point and behaves asymptotically like a logarithmic spiral.

Case II: , . For sufficiently small the image of every trajectory that intersects the disc tends to in one direction and leaves in the other direction. The modulus of varies monotonically on the image of the trajectory in . Different images of trajectories have different limiting directions at the point .

Case III: , . For each there is a number such that for the image of a trajectory that intersects the circle is a Jordan curve lying in the circular annulus .

References

[1]	J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)

Comments

Cf. Quadratic differential for the notion of the trajectory of a quadratic differential.

This description is taken essentially from section 3.2 of [1]. For a detailed treatment of quadratic differentials see also [a1].

For the global structure see Global structure of trajectories.

References

[a1]	K. Strebel, "Quadratic differentials" , Springer (1984) (Translated from German)
[a2]	F.P. Gardiner, "Teichmüller theory and quadratic differentials" , Wiley (1987)