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 $R$ be an oriented Riemann surface and let $Q(z)\,dz^2$ be a quadratic differential on $R$; let $C$ be the set of all zeros and simple poles of $Q(z)\,dz^2$ and let $H$ be the set of all poles of $Q(z)\,dz^2$ of order $\geq2$. The trajectories of $Q(z)\,dz^2$ form a regular family of curves on $R\setminus(C\cup H)$. Under an extension of the concept of a regular family of curves this remains true on $R\setminus H$ also. The behaviour of the trajectories in neighbourhoods of points of $H$ is significantly more complicated. A complete description of the local structure of trajectories is given below.

a) For any point $P\in R\setminus(C\cup H)$ there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ ($w=u+iv$) such that a maximal open arc of each trajectory in $N$ goes to a segment on which $v$ is constant. Consequently, through each point of $R\setminus(C\cup H)$ there passes a trajectory of $Q(z)\,dz^2$ that is either an open arc or a Jordan curve on $R$.

b) For any point $P\in C$ of order $\mu$ ($\mu>0$ if $P$ is a zero and $\mu=-1$ if $P$ is a simple pole) there is a neighbourhood $N$ of $P$ on $R$ and a homeomorphic mapping of $N$ onto the disc $|w|<1$ such that a maximal arc of each trajectory in $N$ goes to an open arc on which $\operatorname{Im}w^{(\mu+2)/2}$ is constant. There are $\mu+2$ trajectories with ends at $P$ and with limiting tangential directions that make equal angles $2\pi/(\mu+2)$ with each other.

c) Let $P\in H$ be a pole of order $\mu>2$. If a certain trajectory has an end at $P$, then it tends to $P$ along one of $\mu-2$ directions making equal angles $2\pi/(\mu-2)$. There is a neighbourhood $N$ of $P$ on $R$ with the following properties: 1) every trajectory that passes through some point of $N$ in each of the directions either tends to $P$ or leaves $N$; 2) there is a neighbourhood $N^*$ of $P$ contained in $N$ and such that every trajectory that passes through some point of $N^*$ tends to $P$ in at least one direction, remaining in $N^*$; 3) if some trajectory lies entirely in $N$ and therefore tends to $P$ in both directions, then the tangent to this trajectory as $P$ is approached in the corresponding direction tends to one of two adjacent limiting positions. The Jordan curve obtained by adjoining $P$ to this trajectory bounds a domain $D$ containing points of the angle formed by the two adjacent limiting tangents. The tangent to any trajectory that has points in common with $D$ tends to these adjacent limiting positions as $P$ is approached in the two directions. By means of a suitable branch of the function $\zeta=\int[Q(z)]^{1/2}\,dz$ the domain $D$ is mapped onto the half-plane $\operatorname{Im}\zeta>c$ (where $c$ 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 $P\in H$ be a pole of order two and let $z$ be the local parameter in terms of which $P$ is the point $z=0$. Suppose that $[Q(z)]^{-1/2}$ has (for some choice of the branch of the root) the following expansion in a neighbourhood of $z=0$:

$$[Q(z)]^{-1/2}=(a+ib)z\{1+b_1z+b_2z^2+\dotsb\},$$

where $a$ and $b$ are real constants and $b_1,b_2,\dots,$ are complex constants. The structure of the images of the trajectories of the differential $Q(z)\,dz^2$ in the $z$-plane is determined by which of the following three cases holds.

Case I: $a\neq0$, $b\neq0$. For sufficiently small $\alpha>0$ the image of each trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction, and leaves $|z|<\alpha$ in the other direction. Both the modulus and the argument of $z$ vary monotonically on the image of the trajectory in $|z|<\alpha$. Each image of a trajectory twists around the point $z=0$ and behaves asymptotically like a logarithmic spiral.

Case II: $a\neq0$, $b=0$. For sufficiently small $\alpha>0$ the image of every trajectory that intersects the disc $|z|<\alpha$ tends to $z=0$ in one direction and leaves $|z|<\alpha$ in the other direction. The modulus of $z$ varies monotonically on the image of the trajectory in $|z|<\alpha$. Different images of trajectories have different limiting directions at the point $z=0$.

Case III: $a=0$, $b\neq0$. For each $\epsilon>0$ there is a number $\alpha(\epsilon)>0$ such that for $0<\alpha\leq\alpha(\epsilon)$ the image of a trajectory that intersects the circle $|z|=\alpha$ is a Jordan curve lying in the circular annulus $\alpha(1+\epsilon)^{-1}<|z|<\alpha(1+\epsilon)$.

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)