Global structure of trajectories
of a quadratic differential
A description of the behaviour as a whole of trajectories of a positive quadratic differential on a compact oriented Riemann surface (cf. Quadratic differential for the definition of trajectory in this setting). Let be a compact oriented Riemann surface, let
be a positive quadratic differential on
, let
be the set of all zeros and simple poles of
, and let
be the set of poles of
of order
. The trajectories of
form a family
which has many of the properties of regular families of curves. This family of curves covers
except for the points of the set
, i.e. through every point of
passes a unique element of
. The behaviour of a trajectory of
in a neighbourhood of any point of
is described by the local structure of the trajectories of the quadratic differential (cf. Local structure of trajectories). In considering the global structure of the curves of
at the points of
, an important role is played by the following unions of trajectories. Let
be the union of all trajectories of
having limit end points at some point of
; let
be the subset of
that is the union of all the trajectories of
which have one limit end point at a point of
and a second limit end point at a point of
.
A set on
is called an
-set with respect to
if each trajectory of
intersecting with
is completely contained in
. The internal closure of the set
is defined as the interior of the closure
and is denoted by
. The internal closure of an
-set is also an
-set. The terminal domain
with respect to
is the largest connected open
-set on
with the following properties: 1)
contains no points of
; 2)
is filled with trajectories of
, each one of which has a limit end point in each one of the two possible directions at a given point
; and 3)
is conformally mapped by the function
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onto the left or right half-plane of the -plane (depending on the choice of the branch of the root). It follows from the local structure of the trajectories of
that the point
should be a pole of the differential
of order at least three.
The strip-like domain with respect to
is the largest connected open
-set on
with the following properties: 1)
contains no points of
; 2)
is filled with the trajectories of
, each one of which has at one point
a limit end point in one direction and at another point
(which may coincide with
) a limit end point in the other direction; and 3)
is conformally mapped by the function
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onto the strip , where
and
are finite real numbers and
. The points
and
may be poles of
of order two or larger.
The circular domain with respect to
is the largest connected open
-set on
with the following properties: 1)
contains a unique double pole
of
; 2)
is filled with the trajectories of
each one of which is a closed Jordan curve which separates
from the boundary of
; and 3) if a purely-imaginary constant
has been suitably chosen, the function
![]() |
supplemented by the value zero at , conformally maps
onto a disc
, and
is mapped to
.
The annular domain with respect to
is the largest connected
-set on
with the following properties: 1)
does not contain any points of
; 2)
is filled with trajectories of
each one of which is a closed Jordan curve; and 3) if a purely-imaginary constant
is suitably chosen, the function
![]() |
conformally maps onto a circular annulus
,
.
The dense domain with respect to
is the largest connected
-set on
with the following properties: 1)
does not contain any points of
; and 2)
is filled with trajectories of
, each one of which is everywhere-dense in
.
The basic structure theorem is valid [2]. Let be a compact oriented Riemann surface and let
be a positive quadratic differential on
, while excluding the following possible cases and all configurations obtainable from such cases by way of a conformal mapping: I.
is a
-sphere,
; II.
is a
-sphere,
,
being positive and
being a real number; and III.
is a torus, and
is regular on
. Then 1)
consists of a finite number of terminal, strip-like, annular, and dense domains; 2) each such domain is bounded by a finite number of trajectories together with points at which the latter meet; each boundary component of such a domain contains a point of
, except for boundary components of the circular or annular domain which may coincide with a boundary components of
; for a strip-like domain two boundary elements issuing from points of the set
subdivide the boundary into two parts, each one of which contains a point of the set
; 3) each pole of
of order
has a neighbourhood that can be covered by the internal closure of the union of
terminal domains and a finite number (which may also be equal to zero) of strip-like domains; and 4) each pole of
of order
has a neighbourhood that can be covered by the internal closure of the union of a finite number of strip-like domains, or has a neighbourhood contained in a circular domain.
The statement of the basic structure theorem of J.A. Jenkins [1] in its original formulation immediately follows from this theorem: Under the conditions of the theorem the set consists of a finite number of terminal, strip-like, circular, and annular domains. In a number of studies in the theory of univalent functions, main stress is laid on proving the fact that the set
is empty for the quadratic differential
under consideration. The search for conditions under which
is empty is also of interest in its own right. The following three-pole theorem provides an example of a quadratic differential
on the
-sphere for which the set
is empty: If
is the
-sphere and
is a quadratic differential on
with at most three different poles, then
is empty.
References
[1] | J.A. Jenkins, "Univalent functions and conformal mappings" , Springer (1958) |
[2] | J.A. Jenkins, "On the global structure of the trajectories of a positive quadratic differential" Illinois J. Math. , 4 : 3 (1960) pp. 405–412 |
Global structure of trajectories. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Global_structure_of_trajectories&oldid=47099