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

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

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