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
:
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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) |
Local structure of trajectories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_structure_of_trajectories&oldid=11783