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on a Riemann surface $R$

A rule which associates to each local parameter $z$ (cf. Local uniformizing parameter) mapping a parametric neighbourhood $U \subset R$ into the extended complex plane $\overline{\mathbf C}$ ($z : U \rightarrow \overline{\mathbf C}$), a function $Q _ {z} : z ( U) \rightarrow \overline{\mathbf C}$ such that for any local parameters $z _ {1} : U _ {1} \rightarrow \overline{\mathbf C}\;$ and $z _ {2} : U _ {2} \rightarrow \overline{\mathbf C}\;$ with $U _ {1} \cap U _ {2}$ non-empty, the following holds in this intersection:

$$\tag{1 } \frac{Q _ {z _ {2} } ( z _ {2} ( p) ) }{Q _ {z _ {1} } ( z _ {1} ( p) ) } = \ \left ( \frac{d z _ {1} ( p) }{d z _ {2} ( p) } \right ) ^ {2} ,\ \ p \in U _ {1} \cap U _ {2} ;$$

here $z ( U)$ is the image of $U$ in $\overline{\mathbf C}$ under $z$. A quadratic differential is often denoted by the symbol $Q ( z ) d z ^ {2}$, to which is attributed the invariance with respect to the choice of the local parameter $z$, as indicated by (1). In other words, a quadratic differential is a non-linear differential of type $( 2 , 0 )$ on a Riemann surface.

The functions $Q _ {z} ( \cdot )$ entering into the definition of a quadratic differential are ordinarily assumed to be measurable or even analytic. In the latter case the quadratic differential is called analytic. A point $p \in R$ is called a zero (or pole) of $Q ( z) d z ^ {2}$ of order $k$ if for each local parameter $z$ the function $Q _ {z} ( \cdot )$ has a zero (or pole) of order $k$ at $p$. The zeros and poles of a quadratic differential are called critical points of it. The zeros and simple poles are called finite critical points and their totality is denoted by $C$. The set of all poles of order $k \geq 2$ is denoted by $H$. If a curve $\gamma \subset R$ has a tangent at each of its points $q$ with respect to the parameter $z$, with tangent vector $a _ {z} ( q)$, and

$$\tag{2 } Q _ {z} ( z ( q) ) ( a _ {z} ( q) ) ^ {2} > 0 ,\ \ q \in \gamma ,$$

then $Q ( z) d z ^ {2}$ is said to be positive, and one writes $Q ( z) d z ^ {2} > 0$, on the curve $\gamma$. If (2) holds with the $>$ sign replaced by $<$, then $Q ( z) d z ^ {2}$ is called negative $( Q ( z) d z ^ {2} < 0 )$ on $\gamma$. Each maximal regular curve on $R$ for which $Q ( z) d z ^ {2} > 0$ (or $Q ( z) d z ^ {2} < 0$) is called a trajectory (or orthogonal trajectory) of the quadratic differential $Q ( z) d z ^ {2}$.

A quadratic differential $Q ( z) d z ^ {2}$ defined on a finite Riemann surface $R$ belongs to $R$ if the boundary $\partial R$ of $R$ is either empty or consists of a finite number of points $p \notin H$ and arcs $\gamma$ on each of which $Q ( z) d z ^ {2}$ is regular and positive or negative. If, furthermore, $\partial R$ is empty or if $Q ( z) d z ^ {2}$ is regular and positive on $\partial R$, then $Q ( z) d z ^ {2}$ is called a positive quadratic differential on the Riemann surface $R$. The metric $| Q ( z) | ^ {1 / 2 } | dz |$, called a $Q$-metric, is single-valued on $R$ and invariant with respect to the choice of the local parameter $z$.

In some neighbourhood $U$ of any point $p \in R \setminus ( C \cup H )$, the function

$$\zeta ( q) = \ \int\limits _ { z( p)} ^ { z( q)} Q ( z) ^ {1/2} d z$$

is regular, single-valued and univalent for each choice of the sign of the integrand; furthermore, each maximal arc of a trajectory (or orthogonal trajectory) of $U$ is converted under $\zeta ( q)$ into a horizontal (or vertical) line interval. Therefore, through each point $p \in R \setminus ( C \cup H )$ passes a trajectory which is either an open arc or a Jordan curve on $R$. The topological and conformal structures of the family of trajectories in a small neighbourhood of each critical point $r$ are completely classified in their dependence on the order of the critical point $r$ and (if $r$ is a pole of the second order and $z ( r) = 0$) on

$$\mathop{\rm arg} \lim\limits _ {q \rightarrow r } Q _ {z} ( z ( q) ) z ( q) ^ {2}$$

(see Local structure of trajectories). A description of the global structure of trajectories is known for finite Riemann surfaces and has many important applications (see also ).

O. Teichmüller has investigated the role of quadratic differentials in the theory of extremal conformal and quasi-conformal mapping and in the solution of moduli problems of Riemann surfaces (see ). He formulated a principle according to which certain quadratic differentials can be associated with extremal problems in the geometric theory of functions, where to each type of extremal problem correspond specific singularities of the quadratic differential (poles), and the geometric properties of the solution are related in a suitable fashion to the structure of the trajectories of the quadratic differential. Inequalities for the coefficients of univalent functions (cf. Univalent function) have been proved in terms of quadratic differentials. A more general inequality for the coefficients of univalent functions in families of domains distributed on a finite Riemann surface is called the general coefficient theorem and is a concrete realization of the Teichmüller principle for a wide class of problems (see , ). The Teichmüller principle also enables one to establish a special coefficient theorem and to solve a large number of concrete extremal problems (see , ).

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