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 $ R $ be a compact oriented Riemann surface, let $ Q( z) d z ^ {2} $ be a positive quadratic differential on $ R $, let $ C $ be the set of all zeros and simple poles of $ Q( z) d z ^ {2} $, and let $ H $ be the set of poles of $ Q( z) d z ^ {2} $ of order $ \geq 2 $. The trajectories of $ Q ( z) d z ^ {2} $ form a family $ F $ which has many of the properties of regular families of curves. This family of curves covers $ R $ except for the points of the set $ C \cup H $, i.e. through every point of $ R \setminus ( C \cup H) $ passes a unique element of $ F $. The behaviour of a trajectory of $ Q( z) d z ^ {2} $ in a neighbourhood of any point of $ R $ 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 $ F $ at the points of $ R \setminus H $, an important role is played by the following unions of trajectories. Let $ \Phi $ be the union of all trajectories of $ Q( z) d z ^ {2} $ having limit end points at some point of $ C $; let $ \Lambda $ be the subset of $ \Phi $ that is the union of all the trajectories of $ Q( z) d z ^ {2} $ which have one limit end point at a point of $ C $ and a second limit end point at a point of $ C \cup H $.

A set $ K $ on $ R $ is called an $ F $- set with respect to $ Q( z) d z ^ {2} $ if each trajectory of $ Q( z) d z ^ {2} $ intersecting with $ K $ is completely contained in $ K $. The internal closure of the set $ K $ is defined as the interior of the closure $ \overline{K}\; $ and is denoted by $ \widehat{K} $. The internal closure of an $ F $- set is also an $ F $- set. The terminal domain $ E $ with respect to $ Q( z) d z ^ {2} $ is the largest connected open $ F $- set on $ R $ with the following properties: 1) $ E $ contains no points of $ C \cup H $; 2) $ E $ is filled with trajectories of $ Q( z) d z ^ {2} $, each one of which has a limit end point in each one of the two possible directions at a given point $ A \in H $; and 3) $ E $ is conformally mapped by the function

$$ \zeta = \int\limits [ Q ( z)] ^ {1/2} dz $$

onto the left or right half-plane of the $ \zeta $- plane (depending on the choice of the branch of the root). It follows from the local structure of the trajectories of $ Q( z) d z ^ {2} $ that the point $ A $ should be a pole of the differential $ Q( z) d z ^ {2} $ of order at least three.

The strip-like domain $ S $ with respect to $ Q( z) d z ^ {2} $ is the largest connected open $ F $- set on $ R $ with the following properties: 1) $ S $ contains no points of $ C \cup H $; 2) $ S $ is filled with the trajectories of $ Q( z) d z ^ {2} $, each one of which has at one point $ A \in H $ a limit end point in one direction and at another point $ B \in H $( which may coincide with $ A $) a limit end point in the other direction; and 3) $ S $ is conformally mapped by the function

$$ \zeta = \int\limits [ Q ( z)] ^ {1/2} dz $$

onto the strip $ a < \mathop{\rm Im} \zeta < b $, where $ a $ and $ b $ are finite real numbers and $ a < b $. The points $ A $ and $ B $ may be poles of $ Q( z) d z ^ {2} $ of order two or larger.

The circular domain $ {\mathcal C} $ with respect to $ Q( z) d z ^ {2} $ is the largest connected open $ F $- set on $ R $ with the following properties: 1) $ {\mathcal C} $ contains a unique double pole $ A $ of $ Q( z) d z ^ {2} $; 2) $ {\mathcal C} \setminus A $ is filled with the trajectories of $ Q( z) d z ^ {2} $ each one of which is a closed Jordan curve which separates $ A $ from the boundary of $ {\mathcal C} $; and 3) if a purely-imaginary constant $ c $ has been suitably chosen, the function

$$ w = \mathop{\rm exp} \left \{ c \int\limits [ Q ( z)] ^ {1/2} dz \right \} , $$

supplemented by the value zero at $ A $, conformally maps $ {\mathcal C} $ onto a disc $ | w | < R $, and $ A $ is mapped to $ w = 0 $.

The annular domain $ D $ with respect to $ Q( z) d z ^ {2} $ is the largest connected $ F $- set on $ R $ with the following properties: 1) $ D $ does not contain any points of $ C \cup H $; 2) $ D $ is filled with trajectories of $ Q( z) d z ^ {2} $ each one of which is a closed Jordan curve; and 3) if a purely-imaginary constant $ c $ is suitably chosen, the function

$$ w = \mathop{\rm exp} \left \{ c \int\limits [ Q ( z)] ^ {1/2} dz \right \} $$

conformally maps $ D $ onto a circular annulus $ r _ {1} < | w | \leq r _ {2} $, $ 0 < r _ {1} < r _ {2} $.

The dense domain $ {\mathcal F} $ with respect to $ Q( z) d z ^ {2} $ is the largest connected $ F $- set on $ R $ with the following properties: 1) $ {\mathcal F} $ does not contain any points of $ H $; and 2) $ {\mathcal F} \setminus C $ is filled with trajectories of $ Q( z) d z ^ {2} $, each one of which is everywhere-dense in $ {\mathcal F} $.

The basic structure theorem is valid [2]. Let $ R $ be a compact oriented Riemann surface and let $ Q( z) d z ^ {2} $ be a positive quadratic differential on $ R $, while excluding the following possible cases and all configurations obtainable from such cases by way of a conformal mapping: I. $ R $ is a $ z $- sphere, $ Q ( z) d z ^ {2} = d z ^ {2} $; II. $ R $ is a $ z $- sphere, $ Q( z) d z ^ {2} = K e ^ {i \alpha } d z / z ^ {2} $, $ K $ being positive and $ \alpha $ being a real number; and III. $ R $ is a torus, and $ Q( z) d z ^ {2} $ is regular on $ R $. Then 1) $ R \setminus \overline \Lambda \; $ 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 $ C $, except for boundary components of the circular or annular domain which may coincide with a boundary components of $ R $; for a strip-like domain two boundary elements issuing from points of the set $ H $ subdivide the boundary into two parts, each one of which contains a point of the set $ C $; 3) each pole of $ Q( z) d z ^ {2} $ of order $ m > 2 $ has a neighbourhood that can be covered by the internal closure of the union of $ m - 2 $ terminal domains and a finite number (which may also be equal to zero) of strip-like domains; and 4) each pole of $ Q( z) d z ^ {2} $ of order $ m = 2 $ 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 $ R \setminus \overline \Phi \; $ 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 $ \widehat \Phi $ is empty for the quadratic differential $ Q( z) d z ^ {2} $ under consideration. The search for conditions under which $ \widehat \Phi $ is empty is also of interest in its own right. The following three-pole theorem provides an example of a quadratic differential $ Q( z) d z ^ {2} $ on the $ z $- sphere for which the set $ \widehat \Phi $ is empty: If $ R $ is the $ z $- sphere and $ Q( z) d z ^ {2} $ is a quadratic differential on $ R $ with at most three different poles, then $ \widehat \Phi $ 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